login
Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
13

%I #44 Sep 08 2022 08:45:11

%S 1,1,1,2,4,2,5,15,15,5,14,56,84,56,14,42,210,420,420,210,42,132,792,

%T 1980,2640,1980,792,132,429,3003,9009,15015,15015,9009,3003,429,1430,

%U 11440,40040,80080,100100,80080,40040,11440,1430,4862,43758,175032,408408,612612,612612,408408,175032,43758,4862

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

%C Coefficients of terms in the series reversion of (1-k*x-(k+1)*x^2)/(1+x). - _Paul Barry_, May 21 2005

%C Equals A131427 * A007318 as infinite lower triangular matrices. [_Philippe Deléham_, Sep 15 2008]

%C 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

%C Diagonal sums are A052709(n+1). - _Philippe Deléham_, Nov 15 2013

%H G. C. Greubel, <a href="/A085880/b085880.txt">Rows n = 0..100 of triangle, flattened</a>

%H Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.

%F 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.

%F Sum_{k>=0} T(n, k) = A151374(n) (row sums). - _Philippe Deléham_, Aug 11 2005

%F G.f.: (1-sqrt(1-4*(x+y)))/(2*(x+y)). - _Vladimir Kruchinin_, Apr 09 2015

%e Triangle starts:

%e [ 1] 1;

%e [ 2] 1, 1;

%e [ 3] 2, 4, 2;

%e [ 4] 5, 15, 15, 5;

%e [ 5] 14, 56, 84, 56, 14;

%e [ 6] 42, 210, 420, 420, 210, 42;

%e [ 7] 132, 792, 1980, 2640, 1980, 792, 132;

%e [ 8] 429, 3003, 9009, 15015, 15015, 9009, 3003, 429;

%e [ 9] 1430, 11440, 40040, 80080, 100100, 80080, 40040, 11440, 1430;

%e [10] 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;

%e ...

%p seq(seq(binomial(n, k)*binomial(2*n, n)/(n+1), k = 0..n), n = 0..10); # _G. C. Greubel_, Feb 07 2020

%t Table[Binomial[n, k]*CatalanNumber[n], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 07 2020 *)

%o (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

%o (Magma) [Binomial(n,k)*Catalan(n): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 07 2020

%o (Sage) [[binomial(n,k)*catalan_number(n) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 07 2020

%o (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

%K nonn,tabl

%O 0,4

%A _N. J. A. Sloane_, Aug 17 2003