Revision History for A209139
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#13 by Giovanni Resta at Fri Jan 24 03:25:13 EST 2020
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#12 by Michel Marcus at Fri Jan 24 00:47:33 EST 2020
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#11 by Jon E. Schoenfield at Thu Jan 23 22:14:14 EST 2020
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#10 by Jon E. Schoenfield at Thu Jan 23 22:14:10 EST 2020
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| COMMENTS
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columnColumn 1: A000045 (Fibonacci numbers)).
alternatingAlternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1...
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| FORMULA
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u(n,x)=) = u(n-1,x)+() + (x+1)*v(n-1,x),
v(n,x)=() = (x+1)*u(n-1,x)+) + 2x*v(n-1,x),
Contribution fromFrom Philippe Deléham, Apr 11 2012. (: (Start)
As DELTA-triangle T(n,k) with 0<= <= k<= <= n ::
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k< < 0 or if k> > n. (End)
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| EXAMPLE
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1;
2..., 1;
3..., 5...., 3;
5..., 12..., 15..., 7;
8..., 27..., 45..., 42..., 17;
First three polynomials u(n,x): 1, 2 + x, 3 + 5x + 3x^2):
1
2 + x
3 + 5x + 3x^2
From Philippe Deléham, Apr 11 2012: (Start)
(1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins ::
1;
1, , 0;
2, , 1, , 0;
3, , 5, , 3, , 0;
5, 12, , 15, , 7, , 0;
8, 27, , 45, , 42, , 17, , 0;
13, 55, 119, 151, 116, 41, 0 . _Philippe Deléham_, Apr 11 2012; (End)
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approved
editing
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#9 by N. J. A. Sloane at Sun Sep 08 19:59:31 EDT 2013
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| COMMENTS
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Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - . - _Philippe Deléham, _, Apr 11 2012
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Contribution from _Philippe Deléham, _, Apr 11 2012. (Start)
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| EXAMPLE
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13, 55, 119, 151, 116, 41, 0 . . _Philippe Deléham, _, Apr 11 2012
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Discussion
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| Sun Sep 08
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| OEIS Server: https://oeis.org/edit/global/1941
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#8 by N. J. A. Sloane at Fri Feb 22 14:40:32 EST 2013
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| COMMENTS
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Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe Deléham, Apr 11 2012
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| FORMULA
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Contribution from DELEHAM Philippe Deléham, Apr 11 2012. (Start)
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| EXAMPLE
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13, 55, 119, 151, 116, 41, 0 . DELEHAM Philippe Deléham, Apr 11 2012
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Discussion
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| Fri Feb 22
| 14:40
| OEIS Server: https://oeis.org/edit/global/1863
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#7 by T. D. Noe at Wed Apr 11 15:35:59 EDT 2012
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#6 by DELEHAM Philippe at Wed Apr 11 14:46:55 EDT 2012
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#5 by DELEHAM Philippe at Wed Apr 11 14:46:45 EDT 2012
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| COMMENTS
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Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - DELEHAM Philippe, Apr 11 2012
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Contribution from DELEHAM Philippe, Apr 11 2012. (Start)
As DELTA-triangle T(n,k) with 0<=k<=n :
G.f.: (1-2*y*x-y*x^2-y^2*x^2)/(1-x-x^2-2*y*x-y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)
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| EXAMPLE
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(1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 5, 3, 0
5, 12, 15, 7, 0
8, 27, 45, 42, 17, 0
13, 55, 119, 151, 116, 41, 0 . DELEHAM Philippe, Apr 11 2012
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approved
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#4 by Russ Cox at Fri Mar 30 18:58:14 EDT 2012
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| AUTHOR
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_Clark Kimberling (ck6(AT)evansville.edu), _, Mar 05 2012
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Discussion
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| Fri Mar 30
| 18:58
| OEIS Server: https://oeis.org/edit/global/285
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