A336169 - OEIS

A336169

Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * multinomial(n+(k-1)*j; n-j, {j}^k).

1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 5, 1, 0, 1, 1, 23, 67, 1, 0, 0, 1, 119, 2401, 1109, 1, 0, 1, 1, 719, 112681, 347279, 20251, 1, 0, 0, 1, 5039, 7479361, 166923119, 58370761, 391355, 1, 0, 1, 1, 40319, 681040081, 137127810959, 302857024681, 10693893503, 7847155, 1, 0, 0, 1, 362879, 81729285121, 182499151015439, 3244063941457921, 616967236620839, 2071837562929, 161476565, 1, 0, 1

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OFFSET

0,12

COMMENTS

Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j + Product_{j=1..k} x_j) for k>0.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1+x)^(k*j+1).

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, ...

0, 0, 1, 5, 23, 119, ...

1, 0, 1, 67, 2401, 112681, ...

0, 0, 1, 1109, 347279, 166923119, ...

1, 0, 1, 20251, 58370761, 302857024681, ...

0, 0, 1, 391355, 10693893503, 616967236620839, ...

MATHEMATICA

T[n_, k_] := Sum[(-1)^(n - j)*(n + (k - 1)*j)!/(n - j)!/(j!)^k, {j, 0, n} ]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 10 2020 *)

CROSSREFS

Columns k=0-5 give: A059841, A000007, A000012, A124435, A336170, A336171.

Rows n=0-1 give: A000012, A033312.

Main diagonal gives A336172.

Cf. A229142.

Sequence in context: A293087 A209349 A242095 * A007912 A019755 A085475

Adjacent sequences: A336166 A336167 A336168 * A336170 A336171 A336172

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Jul 10 2020

STATUS

approved