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

0,4

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

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.

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

Example

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;
...

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 *)

Prog

(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

Crossrefs

Sequence in context: A167685 A268740 A120493 * A055883 A366588 A085843

Adjacent sequences: A085877 A085878 A085879 * A085881 A085882 A085883

Keyword

nonn,tabl

Author

N. J. A. Sloane, Aug 17 2003

Status

approved