A279637 -id:A279637

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A279636

Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

+10
12

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 5, 10, 15, 1, 1, 9, 22, 41, 52, 1, 1, 17, 52, 125, 196, 203, 1, 1, 33, 130, 413, 836, 1057, 877, 1, 1, 65, 340, 1445, 3916, 6277, 6322, 4140, 1, 1, 129, 922, 5261, 19676, 41077, 52396, 41393, 21147, 1, 1, 257, 2572, 19685, 104116, 288517, 481384, 479593, 293608, 115975

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

OFFSET

0,6

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Wikipedia, Kronecker delta

FORMULA

E.g.f. of column k: exp(exp(x)*(Sum_{j=0..k} Stirling2(n,j)*x^j) - delta_{0,k}).

EXAMPLE

Square array A(n,k) begins:

: 1, 1, 1, 1, 1, 1, 1, ...

: 2, 3, 5, 9, 17, 33, 65, ...

: 5, 10, 22, 52, 130, 340, 922, ...

: 15, 41, 125, 413, 1445, 5261, 19685, ...

: 52, 196, 836, 3916, 19676, 104116, 572036, ...

: 203, 1057, 6277, 41077, 288517, 2133397, 16379797, ...

MAPLE

egf:= k-> exp(exp(x)*add(Stirling2(k, j)*x^j, j=0..k)-`if`(k=0, 1, 0)):

A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):

seq(seq(A(n, d-n), n=0..d), d=0..12);

# second Maple program:

A:= proc(n, k) option remember; `if`(n=0, 1,

add(binomial(n-1, j-1)*j^k*A(n-j, k), j=1..n))

end:

seq(seq(A(n, d-n), n=0..d), d=0..12);

MATHEMATICA

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n-1, j-1]*j^k*A[n-j, k], {j, 1, n}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000110, A000248, A033462, A279358, A279637, A279638, A279639, A279640, A279641, A279642, A279643.

Rows n=0+1,2 give: A000012, A000051.

Main diagonal gives A279644.

Cf. A145460.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 16 2016

STATUS

approved

A306325

Expansion of e.g.f. log(1 + exp(x)*x*(1 + 7*x + 6*x^2 + x^3)).

+10
2

0, 1, 15, 35, -650, -5251, 83376, 1623439, -19261584, -836109351, 5365104400, 636771444011, 561938325312, -661384866976523, -7128491581221360, 879709224738485415, 21742632225425026816, -1413667730904479933647, -64871991410092201623024, 2556051301724027073500035, 212244727356899863738042560

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

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..384

FORMULA

E.g.f.: log(1 + Sum_{k>=1} k^4*x^k/k!).

a(0) = 0; a(n) = n^4 - (1/n)*Sum_{k=1..n-1} binomial(n,k)*(n - k)^4*k*a(k).

MAPLE

a:=series(log(1 + exp(x)*x*(1 + 7*x + 6*x^2 + x^3)), x=0, 21): seq(n!*coeff(a, x, n), n=0..20); # Paolo P. Lava, Mar 26 2019

MATHEMATICA

nmax = 20; CoefficientList[Series[Log[1 + Exp[x] x (1 + 7 x + 6 x^2 + x^3)], {x, 0, nmax}], x] Range[0, nmax]!

a[n_] := a[n] = n^4 - Sum[Binomial[n, k] (n - k)^4 k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 20}]

CROSSREFS

Cf. A000583, A033464, A279637, A300452.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Feb 07 2019

STATUS

approved

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