%I #22 Mar 01 2021 17:53:47

%S 1,1,3,1,16,13,1,125,171,39,1,1296,2551,1091,101,1,16807,43653,28838,

%T 5498,243,1,262144,850809,780585,243790,24270,561,1,4782969,18689527,

%U 22278189,10073955,1733035,98661,1263,1

%N Triangle T(n,k) = ((-1)^(n+k)/(n+1))*Sum_{j=0..n} (-1)^j*j!*Stirling2(n, j)* binomial(n-j, k)*binomial(n+j, j), read by rows.

%C Row sums are A001761.

%H G. C. Greubel, <a href="/A156653/b156653.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, m) = [x^m] p(x,n) where p(x,n) = (1-x)^(2*n+1)/((n+1)*x^n)*Sum_{k>=0} (k+1)^n* binomial(k, n)*x^k.

%F T(n, m) = 1/(n+1)*Sum_{k=0..n} (-1)^(n+m+k)*k!*Stirling2(n,k)*C(n-k,m)*C(n+k,k). - _Vladimir Kruchinin_, May 05 2020

%F E.g.f. satisfies: A(x,y) = x*E(A(x,y),y), where E(x,y) is e.g.f. of Euler numbers of first kind A008292. - _Vladimir Kruchinin_, May 05 2020

%e Triangle begins as:

%e 1;

%e 1;

%e 3, 1;

%e 16, 13, 1;

%e 125, 171, 39, 1;

%e 1296, 2551, 1091, 101, 1;

%e 16807, 43653, 28838, 5498, 243, 1;

%e 262144, 850809, 780585, 243790, 24270, 561, 1;

%e 4782969, 18689527, 22278189, 10073955, 1733035, 98661, 1263, 1;

%e 100000000, 457947691, 677785807, 410994583, 106215619, 10996369, 379693, 2797, 1;

%t T[n_, m_]:= Sum[(-1)^(n+m+k) k! StirlingS2[n, k] Binomial[n-k, m] Binomial[n+k, k], {k, 0, n}]/(n+1);

%t Prepend[Table[T[n, m], {n,10}, {m, 0, n-1}]//Flatten, 1] (* _Peter Luschny_, May 11 2020 *)

%o (Maxima)

%o T(n,m):=sum(k!*stirling2(n,k)*(-1)^(n+m+k)*binomial(n-k,m)*binomial(n+k,k),k,0,n) /(n+1); /* _Vladimir Kruchinin_, May 11 2020 */

%o (Magma)

%o A156653:= func< n,k | ((-1)^(n+k)/(n+1))*(&+[ (-1)^j*Factorial(j)*StirlingSecond(n, j)*Binomial(n-j, k)*Binomial(n+j, j) : j in [0..n]]) >;

%o [1] cat [A156653(n,k): k in [0..n-1], n in [1..12]]; // _G. C. Greubel_, Mar 01 2021

%o (Sage)

%o def A156653(n,k): return ((-1)^(n+k)/(n+1))*sum( (-1)^j*factorial(j)* stirling_number2(n, j)*binomial(n-j, k)*binomial(n+j, j) for j in (0..n))

%o [1]+flatten([[A156653(n,k) for k in (0..n-1)] for n in (1..12)]) # _G. C. Greubel_, Mar 01 2021

%Y Cf. A001761, A008292, A048993.

%K nonn,tabf

%O 0,3

%A _Roger L. Bagula_, Feb 12 2009

%E New name by _Vladimir Kruchinin_, May 11 2020