A156653 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.
1, 1, 3, 1, 16, 13, 1, 125, 171, 39, 1, 1296, 2551, 1091, 101, 1, 16807, 43653, 28838, 5498, 243, 1, 262144, 850809, 780585, 243790, 24270, 561, 1, 4782969, 18689527, 22278189, 10073955, 1733035, 98661, 1263, 1
Offset: 0
Examples
Triangle begins as:
1;
1;
3, 1;
16, 13, 1;
125, 171, 39, 1;
1296, 2551, 1091, 101, 1;
16807, 43653, 28838, 5498, 243, 1;
262144, 850809, 780585, 243790, 24270, 561, 1;
4782969, 18689527, 22278189, 10073955, 1733035, 98661, 1263, 1;
100000000, 457947691, 677785807, 410994583, 106215619, 10996369, 379693, 2797, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Magma
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]]) >; [1] cat [A156653(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Mar 01 2021
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Mathematica
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); Prepend[Table[T[n, m], {n,10}, {m, 0, n-1}]//Flatten, 1] (* Peter Luschny, May 11 2020 *) -
Maxima
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 */
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Sage
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)) [1]+flatten([[A156653(n,k) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Mar 01 2021
Formula
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.
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
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
Extensions
New name by Vladimir Kruchinin, May 11 2020
Comments