A085880 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

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

Offset: 0

Author

N. J. A. Sloane, Aug 17 2003

Keywords

• nonn
• tabl

Comments

Coefficients of terms in the series reversion of (1-k*x-(k+1)*x^2)/(1+x). - Paul Barry, May 21 2005

Equals A131427 * A007318 as infinite lower triangular matrices. [Philippe Deléham, Sep 15 2008]

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

Diagonal sums are A052709(n+1). - Philippe Deléham, Nov 15 2013

Examples

			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;
...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.

Programs

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

Magma

[Binomial(n,k)*Catalan(n): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 07 2020

Maple

seq(seq(binomial(n, k)*binomial(2*n, n)/(n+1), k = 0..n), n = 0..10); # G. C. Greubel, Feb 07 2020

Mathematica

Table[Binomial[n, k]*CatalanNumber[n], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2020 *)

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

Sage

[[binomial(n,k)*catalan_number(n) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 07 2020

Formula

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.

Sum_{k>=0} T(n, k) = A151374(n) (row sums). - Philippe Deléham, Aug 11 2005

G.f.: (1-sqrt(1-4*(x+y)))/(2*(x+y)). - Vladimir Kruchinin, Apr 09 2015