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

0,3

Comments

Row sums are A001761.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Example

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;

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 *)

Prog

(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 */
(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
(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

Crossrefs

Cf. A001761, A008292, A048993.

Sequence in context: A038675 A264902 A350446 * A048159 A276640 A123527

Adjacent sequences: A156650 A156651 A156652 * A156654 A156655 A156656

Keyword

nonn,tabf

Author

Roger L. Bagula, Feb 12 2009

Extensions

New name by Vladimir Kruchinin, May 11 2020

Status

approved