A108953 -id:A108953 - cp's OEIS frontend

A107713 Convolution of 2^n*n! and n!.

1, 3, 12, 66, 484, 4536, 52128, 709776, 11153376, 198339840, 3932962560, 85976743680, 2053285148160, 53173906652160, 1483987541299200, 44396218792396800, 1417294759310438400, 48088097391133900800, 1728013936221838540800, 65558270633421791232000

Offset: 0

Mike Zabrocki, Jun 10 2005

nonn

E.g.f. is int( 1/((1-t)(1-2*(x+t))), t=0..x).

			a(4) = 484 = 4! 0! + 2 3! 1! + 2^2 2! 2! + 2^3 1! 3! + 2^4 0! 4!

Cf. A003149, A108953, A110467.

f:=proc(n) local k; add(2^k*k!*(n-k)!,k=0..n); end:

Rest[Range[0, 20]! CoefficientList[Series[((Log[1 - x] + Log[1 - 2 x]))/(-3 + 2 x), {x, 0, 20}], x]] (* Vincenzo Librandi, Jul 13 2015 *)
Table[Sum[2^k * k! * (n-k)!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 08 2019 *)

a(n) = Sum_{k=0..n} 2^k * k! * (n-k)!.
E.g.f. (for offset 1): (log(1-x)+log(1-2*x))/(-3+2*x).
a(n) ~ n! * 2^n * (1 + 1/(2*n) + 1/(2*n^2) + 5/(4*n^3) + 17/(4*n^4) + 37/(2*n^5) + 98/n^6 + 4885/(8*n^7) + 34969/(8*n^8) + 70657/(2*n^9) + 636151/(2*n^10) + ...). - Vaclav Kotesovec, Aug 08 2019, extended Dec 07 2020

A110467 Convolution of 4^n*n! and n!.

Original entry on oeis.org

1, 5, 38, 430, 6640, 130200, 3088560, 85828080, 2731899456, 97956720000, 3906077932800, 171436911264000, 8211994618982400, 426284974571904000, 23836815193556736000, 1428394963614554880000, 91316330157374106624000

Offset: 0

Paul Barry, Jul 21 2005

G. C. Greubel, Table of n, a(n) for n = 0..360

Cf. A107713, A108953.

Table[Sum[k!*4^k*(n - k)!, {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 28 2017 *)

for(n=0,50, print1(sum(k=0,n, k!*4^k*(n-k)!), ", ")) \\ G. C. Greubel, Aug 28 2017

E.g.f. (for offset 1): log((1-x)*(1-4*x))/(4*x-5).
a(n) = n!*Sum_{k=0..n} 4^k/binomial(n, k).
a(n) = Sum_{k=0..n} k!*4^k*(n-k)!.
a(n) ~ 4^n * n! * (1 + 1/(4*n) + 1/(8*n^2) + 7/(32*n^3) + 1/(2*n^4) + 187/(128*n^5) + 1337/(256*n^6) + 22559/(1024*n^7) + 109517/(1024*n^8) + 1202047/(2048*n^9) + 14710847/(4096*n^10) + ...). - Vaclav Kotesovec, Dec 07 2020

A300519 Convolution of n! and n^n.

Original entry on oeis.org

1, 2, 7, 39, 321, 3603, 51391, 884873, 17770445, 406673247, 10431884283, 296262164637, 9224841015745, 312441152401067, 11434829066996087, 449675059390576257, 18908960744072894325, 846638474386244188311, 40213487658138717885907, 2019543479160709325145893

Offset: 0

Vaclav Kotesovec, Mar 08 2018

nonn

Cf. A000142, A000312, A061711, A277506.

Cf. A107713, A108953, A110467.

Table[Sum[If[k == 0, 1, k^k] * (n-k)!, {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} k^k * (n-k)!.

a(n) ~ n^n * (1 + exp(-1)/n).

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A107713 Convolution of 2^n*n! and n!.

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A110467 Convolution of 4^n*n! and n!.

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A300519 Convolution of n! and n^n.

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