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A085880
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Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
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13
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1, 1, 1, 2, 4, 2, 5, 15, 15, 5, 14, 56, 84, 56, 14, 42, 210, 420, 420, 210, 42, 132, 792, 1980, 2640, 1980, 792, 132, 429, 3003, 9009, 15015, 15015, 9009, 3003, 429, 1430, 11440, 40040, 80080, 100100, 80080, 40040, 11440, 1430, 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862
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OFFSET
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0,4
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COMMENTS
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Coefficients of terms in the series reversion of (1-k*x-(k+1)*x^2)/(1+x). - Paul Barry, May 21 2005
Sum_{k=0..n} T(n,k)*x^k = A168491(n), A000007(n), A000108(n), A151374(n), A005159(n), A151403(n), A156058(n), A156128(n), A156266(n), A156270(n), A156273(n), A156275(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Nov 15 2013
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LINKS
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FORMULA
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Triangle given by [1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
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EXAMPLE
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Triangle starts:
[ 1] 1;
[ 2] 1, 1;
[ 3] 2, 4, 2;
[ 4] 5, 15, 15, 5;
[ 5] 14, 56, 84, 56, 14;
[ 6] 42, 210, 420, 420, 210, 42;
[ 7] 132, 792, 1980, 2640, 1980, 792, 132;
[ 8] 429, 3003, 9009, 15015, 15015, 9009, 3003, 429;
[ 9] 1430, 11440, 40040, 80080, 100100, 80080, 40040, 11440, 1430;
[10] 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;
...
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MAPLE
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seq(seq(binomial(n, k)*binomial(2*n, n)/(n+1), k = 0..n), n = 0..10); # G. C. Greubel, Feb 07 2020
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MATHEMATICA
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Table[Binomial[n, k]*CatalanNumber[n], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 07 2020 *)
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PROG
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(PARI) tabl(nn) = {for (n=0, nn, c = binomial(2*n, n)/(n+1); for (k=0, n, print1(c*binomial(n, k), ", "); ); print(); ); } \\ Michel Marcus, Apr 09 2015
(Magma) [Binomial(n, k)*Catalan(n): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 07 2020
(Sage) [[binomial(n, k)*catalan_number(n) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 07 2020
(GAP) Flat(List([0..10], n-> List([0..n], k-> Binomial(n, k)*Binomial(2*n, n)/( n+1) ))); # G. C. Greubel, Feb 07 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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