A341014 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * j! * binomial(n,j)^2.

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 17, 34, 1, 1, 5, 31, 139, 209, 1, 1, 6, 49, 352, 1473, 1546, 1, 1, 7, 71, 709, 5233, 19091, 13327, 1, 1, 8, 97, 1246, 13505, 95836, 291793, 130922, 1, 1, 9, 127, 1999, 28881, 318181, 2080999, 5129307, 1441729, 1

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

E.g.f. of column k: exp(x/(1-k*x)) / (1-k*x).

T(n,k) = (2*k*n-k+1) * T(n-1,k) - k^2 * (n-1)^2 * T(n-2,k) for n > 1.

Example

Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     3,     4,      5,      6, ...
  1,    7,    17,    31,     49,     71, ...
  1,   34,   139,   352,    709,   1246, ...
  1,  209,  1473,  5233,  13505,  28881, ...
  1, 1546, 19091, 95836, 318181, 830126, ...

Mathematica

T[n_, k_] := Sum[If[j == k == 0, 1, k^j]*j!*Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 02 2021 *)

Prog

(PARI) {T(n, k) = sum(j=0, n, k^j*j!*binomial(n, j)^2)}

Crossrefs

Columns 0..4 give A000012, A002720, A025167, A102757, A102773.

Rows 0..2 give A000012, A000027(n+1), A056220(n+1).

Main diagonal gives A330260.

Cf. A307883.

Sequence in context: A025243 A352765 A361616 * A145085 A228904 A144512

Adjacent sequences: A341011 A341012 A341013 * A341015 A341016 A341017

Keyword

nonn,tabl

Author

Seiichi Manyama, Feb 02 2021

Status

approved