A341033 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-k*x)).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 37, 73, 1, 1, 1, 9, 73, 361, 501, 1, 1, 1, 11, 121, 1009, 4361, 4051, 1, 1, 1, 13, 181, 2161, 17341, 62701, 37633, 1, 1, 1, 15, 253, 3961, 48081, 355951, 1044205, 394353, 1

Offset

0,9

Links

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

Formula

T(n,k) = Sum_{j=1..n} k^(n-j) * (n!/j!) * binomial(n-1,j-1) for n > 0.

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

Example

Square array begins:
  1,   1,    1,     1,     1,      1, ...
  1,   1,    1,     1,     1,      1, ...
  1,   3,    5,     7,     9,     11, ...
  1,  13,   37,    73,   121,    181, ...
  1,  73,  361,  1009,  2161,   3961, ...
  1, 501, 4361, 17341, 48081, 108101, ...

Mathematica

T[0, k_] = 1; T[n_, k_] := n!*Sum[If[k == n - j == 0, 1, k^(n - j)]*Binomial[n - 1, j - 1]/j!, {j, 1, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 03 2021 *)

Prog

(PARI) {T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^(n-j)*binomial(n-1, j-1)/j!))}
(PARI) {T(n, k) = if(n<2, 1, (2*k*n-2*k+1)*T(n-1, k)-k^2*(n-1)*(n-2)*T(n-2, k))}

Crossrefs

Columns 0..7 give A000012, A000262, A025168, A321837, A321847, A321848, A321849, A321850.

Main diagonal gives A293146.

Cf. A253286, A341014.

Sequence in context: A361277 A300853 A293012 * A348481 A274391 A368862

Adjacent sequences: A341030 A341031 A341032 * A341034 A341035 A341036

Keyword

nonn,tabl

Author

Seiichi Manyama, Feb 03 2021

Status

approved