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

%I A085880 #44 Sep 08 2022 08:45:11
%S A085880 1,1,1,2,4,2,5,15,15,5,14,56,84,56,14,42,210,420,420,210,42,132,792,
%T A085880 1980,2640,1980,792,132,429,3003,9009,15015,15015,9009,3003,429,1430,
%U A085880 11440,40040,80080,100100,80080,40040,11440,1430,4862,43758,175032,408408,612612,612612,408408,175032,43758,4862
%N A085880 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
%C A085880 Coefficients of terms in the series reversion of (1-k*x-(k+1)*x^2)/(1+x). - _Paul Barry_, May 21 2005
%C A085880 Equals A131427 * A007318 as infinite lower triangular matrices. [_Philippe Deléham_, Sep 15 2008]
%C A085880 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 A085880 Diagonal sums are A052709(n+1). - _Philippe Deléham_, Nov 15 2013
%H A085880 G. C. Greubel, <a href="/A085880/b085880.txt">Rows n = 0..100 of triangle, flattened</a>
%H A085880 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 A085880 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 A085880 Sum_{k>=0} T(n, k) = A151374(n) (row sums). - _Philippe Deléham_, Aug 11 2005
%F A085880 G.f.: (1-sqrt(1-4*(x+y)))/(2*(x+y)). - _Vladimir Kruchinin_, Apr 09 2015
%e A085880 Triangle starts:
%e A085880 [ 1]     1;
%e A085880 [ 2]     1,     1;
%e A085880 [ 3]     2,     4,      2;
%e A085880 [ 4]     5,    15,     15,      5;
%e A085880 [ 5]    14,    56,     84,     56,     14;
%e A085880 [ 6]    42,   210,    420,    420,    210,     42;
%e A085880 [ 7]   132,   792,   1980,   2640,   1980,    792,    132;
%e A085880 [ 8]   429,  3003,   9009,  15015,  15015,   9009,   3003,    429;
%e A085880 [ 9]  1430, 11440,  40040,  80080, 100100,  80080,  40040,  11440,  1430;
%e A085880 [10]  4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;
%e A085880 ...
%p A085880 seq(seq(binomial(n, k)*binomial(2*n, n)/(n+1), k = 0..n), n = 0..10); # _G. C. Greubel_, Feb 07 2020
%t A085880 Table[Binomial[n, k]*CatalanNumber[n], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 07 2020 *)
%o A085880 (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 A085880 (Magma) [Binomial(n,k)*Catalan(n): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 07 2020
%o A085880 (Sage) [[binomial(n,k)*catalan_number(n) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 07 2020
%o A085880 (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 A085880 nonn,tabl
%O A085880 0,4
%A A085880 _N. J. A. Sloane_, Aug 17 2003