A158777 -id:A158777 - cp's OEIS frontend

A334157 Row sums of array A158777.

1, 2, 5, 16, 89, 686, 5917, 54860, 588401, 7370074, 103522421, 1573237832, 25869057865, 462768222086, 8965777751309, 186025937645956, 4106375449878497, 96241703493486770, 2390797380938894821, 62730027061416412544

Offset: 0

Petros Hadjicostas, Apr 16 2020

nonn

Cf. A003269, A087208, A158777, A330044, A330045.

W := proc(n, m) local v, s, h; v := 0;
for s from 0 to m do
if 0 = (m - s) mod 4 then
h := (m - s)/4;
v := v + binomial(n - s - 3*h, h)/s!;
end if; end do; n!*v; end proc;
seq(add(W(n1, m1), m1 = 0 .. n1), n1 = 0 .. 35);

Table[Apply[Plus, CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/( 1 - x/t - t^4*x^4), {x, 0, 50}], n]], t]], {n, 0, 40}]; (* Program due to Roger L. Bagula from A158777 *)

a(n) = n!*Sum_{k=0..n} A003269(k+1)/(n-k)!.
a(n) = n!*Sum_{k=0..n} Sum_{s=0..floor(k/3)} binomial(k-3*s, s)/(n-k)!.
E.g.f.: exp(x)/(1 - x - x^4).

A330045 Expansion of e.g.f. exp(x) / (1 - x^4).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 841, 42001, 365905, 1819441, 6660721, 498971881, 6278929801, 43710250585, 218205219961, 21795091762081, 358652470233121, 3210080802962401, 20298322381652065, 2534333270094778681, 51516840824285500441, 563561785768079119561

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000522, A087208, A100733, A330044, A334157.

Outer diagonal of A158777.

nmax = 22; CoefficientList[Series[Exp[x]/(1 - x^4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[n!/(n - 4 k)!, {k, 0, Floor[n/4]}], {n, 0, 22}]

G.f.: Sum_{k>=0} (4*k)! * x^(4*k) / (1 - x)^(4*k + 1).
a(0) = a(1) = a(2) = a(3) = 1; a(n) = n*(n - 1)*(n - 2)*(n - 3)*a(n - 4) + 1.
a(n) = Sum_{k=0..floor(n/4)} n! / (n - 4*k)!.
a(n) ~ n! * (2*cos(Pi*n/2 - 1) + exp(1) + (-1)^n*exp(-1))/4. - Vaclav Kotesovec, Apr 18 2020

cp's OEIS Frontend

A334157 Row sums of array A158777.

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