A293012 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1 - x)^k).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 31, 73, 1, 1, 1, 9, 55, 241, 501, 1, 1, 1, 11, 85, 529, 2261, 4051, 1, 1, 1, 13, 121, 961, 6121, 24781, 37633, 1, 1, 1, 15, 163, 1561, 13041, 82711, 309835, 394353, 1, 1, 1, 17, 211, 2353, 24101, 207001, 1273567, 4342241, 4596553, 1

Offset

0,9

Links

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

Formula

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

From Seiichi Manyama, Oct 21 2017: (Start)

Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0. (End)

A(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j-1,n-j)/j!. - Seiichi Manyama, Mar 06 2023

Example

E.g.f. of column k: A_k(x) =  1 + x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 9*k + 1)*x^3/3! + (4*k^3 + 36*k^2 + 32*k + 1)*x^4/4! + ...
Square array begins:
  1,   1,    1,    1,     1,     1,  ...
  1,   1,    1,    1,     1,     1,  ...
  1,   3,    5,    7,     9,    11,  ...
  1,  13,   31,   55,    85,   121,  ...
  1,  73,  241,  529,   961,  1561,  ...
  1, 501, 2261, 6121, 13041, 24101,  ...

Mathematica

Table[Function[k, n! SeriesCoefficient[Exp[x/(1 - x)^k], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Prog

(PARI) T(n, k) = n!*sum(j=0, n, binomial(n+(k-1)*j-1, n-j)/j!); \\ Seiichi Manyama, Mar 06 2023

Crossrefs

Columns k=0..4 give A000012, A000262, A082579, A091695, A361283.

Main diagonal gives A293013.

Sequence in context: A307855 A361277 A300853 * A341033 A348481 A274391

Adjacent sequences: A293009 A293010 A293011 * A293013 A293014 A293015

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Sep 28 2017

Status

approved