Revision History for A154844
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#18 by Charles R Greathouse IV at Thu Sep 08 08:45:40 EDT 2022
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| PROG
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(MAGMAMagma) [[StirlingSecond(n, k) + StirlingSecond(n, n-k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 01 2019
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Discussion
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| Thu Sep 08
| 08:45
| OEIS Server: https://oeis.org/edit/global/2944
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#17 by Bruno Berselli at Thu May 09 04:45:47 EDT 2019
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#16 by Michel Marcus at Wed May 01 02:29:22 EDT 2019
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#15 by Michel Marcus at Wed May 01 02:29:10 EDT 2019
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| NAME
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Triangular sequence of Stirling numbers (A048993) of the secondTriangle kind: T(n, mk) = S(n, mk) + S(n, n-mk), where S are the Stirling numbers (A048993) of the second kind, read by rows.
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| STATUS
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proposed
editing
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Discussion
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| Wed May 01
| 02:29
| Michel Marcus: ok ?
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#14 by G. C. Greubel at Wed May 01 02:21:01 EDT 2019
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#13 by G. C. Greubel at Wed May 01 02:20:38 EDT 2019
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| NAME
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Symmetrical triangularTriangular sequence of Stirling numbers (A048993) of the second kind: tT(n,, m)=StirlingS2[) = S(n, m] + StirlingS2[) + S(n, n - -m].), read by rows.
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| DATA
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2, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 14, 7, 1, 1, 11, 40, 40, 11, 1, 1, 16, 96, 180, 96, 16, 1, 1, 22, 203, 651, 651, 203, 22, 1, 1, 29, 393, 2016, 3402, 2016, 393, 29, 1, 1, 37, 717, 5671, 14721, 14721, 5671, 717, 37, 1, 1, 46, 1261, 15210, 56932, 85050, 56932, 15210, 1261, 46, 1
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| COMMENTS
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Row sums are:: {2, 2, 4, 10, 30, 104, 406, 1754, 8280, 42294, 231950, ...}.
{2, 2, 4, 10, 30, 104, 406, 1754, 8280, 42294, 231950,...}
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| LINKS
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G. C. Greubel, <a href="/A154844/b154844.txt">Rows n = 0..100 of triangle, flattened</a>
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| FORMULA
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tT(n,, m)=StirlingS2[) = S(n, m] + StirlingS2[) + S(n, n - -m].), where S(n,k) = A048993(n,k).
Sum_{k, =0<=k<=..n} T(n,k) = 2*A000110(n). - Philippe Deléham, Feb 17 2013
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| EXAMPLE
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{2},
Triangle begins as:
2;
{ 1, , 1},;
{ 1, , 2, , 1},;
{ 1, , 4, , 4, , 1},;
{ 1, , 7, , 14, , 7, , 1},;
{ 1, 11, , 40, , 40, , 11, , 1},;
{ 1, 16, , 96, , 180, , 96, , 16, , 1},;
{ 1, 22, , 203, , 651, , 651, , 203, , 22, , 1},;
{ 1, 29, , 393, , 2016, , 3402, , 2016, , 393, , 29, , 1},;
{ 1, 37, , 717, , 5671, 14721, 14721, , 5671, , 717, , 37, , 1},;
{ 1, 46, 1261, 15210, 56932, 85050, 56932, 15210, 1261, 46, 1};
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| MATHEMATICA
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Clear[t, n, m]; tTable[n_, m_] = StirlingS2[n, m] + StirlingS2[n, n-m], {n - , 0, 10}, {m]; , 0, n}]//Flatten
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]
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| PROG
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(PARI) {T(n, m) = stirling(n, k, 2) + stirling(n, n-m, 2)}; \\ G. C. Greubel, May 01 2019
(MAGMA) [[StirlingSecond(n, k) + StirlingSecond(n, n-k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 01 2019
(Sage) [[stirling_number2(n, k) + stirling_number2(n, n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 01 2019
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| CROSSREFS
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Cf. A048993.
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| KEYWORD
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nonn,tabl,uned
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| STATUS
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approved
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#12 by N. J. A. Sloane at Wed Sep 16 19:34:19 EDT 2015
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#11 by N. J. A. Sloane at Wed Sep 16 19:34:17 EDT 2015
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| KEYWORD
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nonn,tabl,uned,tabl
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| STATUS
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approved
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#10 by N. J. A. Sloane at Fri Feb 22 14:39:34 EST 2013
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| FORMULA
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Sum_{k, 0<=k<=n} T(n,k) = 2*A000110(n). - _DELEHAM Philippe Deléham_, Feb 17 2013
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Discussion
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| Fri Feb 22
| 14:39
| OEIS Server: https://oeis.org/edit/global/1863
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#9 by Bruno Berselli at Mon Feb 18 06:04:32 EST 2013
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