A351421 -id:A351421

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A351420

Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. -log(1 - f^(k-1)(x)), where f(x) = log(1+x).

+10
6

1, 1, 1, 1, 0, 2, 1, -1, 1, 6, 1, -2, 3, -1, 24, 1, -3, 8, -13, 8, 120, 1, -4, 16, -48, 77, -26, 720, 1, -5, 27, -124, 386, -576, 194, 5040, 1, -6, 41, -259, 1270, -3905, 5219, -1142, 40320, 1, -7, 58, -471, 3244, -16243, 47701, -55567, 9736, 362880

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Table of n, a(n) for n=1..55.

FORMULA

T(n,k) = Sum_{j=1..n} Stirling1(n,j) * T(j,k-1), k>1, T(n,1) = (n-1)!.

EXAMPLE

Square array begins:

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

1, 0, -1, -2, -3, -4, ...

2, 1, 3, 8, 16, 27, ...

6, -1, -13, -48, -124, -259, ...

24, 8, 77, 386, 1270, 3244, ...

120, -26, -576, -3905, -16243, -50375, ...

MATHEMATICA

T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Feb 11 2022 *)

PROG

(PARI) T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));

CROSSREFS

Columns k=1..5 give A000142(n-1), (-1)^(n-1) * A089064(n), A351421, A351422, A351423.

Main diagonal gives A351424.

Cf. A111933, A351429.

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Feb 11 2022

STATUS

approved

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