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A193520 a(n) = Sum_{k=0..n} G(n)/(G(k)*G(n-k)) where G(n) = Product_{k=0..n} k!. 4
1, 2, 4, 14, 122, 3122, 260642, 76214882, 85552669442, 381014246511362, 7442029915221081602, 632869669701185574873602, 264542347321693265938488883202, 517169258108069965039831739271321602, 5495073385198979486456081260457854269542402 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..14.

FORMULA

G.f.: A(x) = ( Sum_{n>=0} x^n/G(n) )^2 where A(x) = Sum_{n>=0} a(n)*x^n/G(n), and G(n) = Product_{k=0..n} k!.

a(n) ~ 2^(n^2/4 + n - 5*(-1)^n/8 + 23/24) * n^(n^2/4 + (-1)^n/8 - 13/24) / (sqrt(Pi) * exp(3*n^2/8 + Zeta'(-1))). - Vaclav Kotesovec, Mar 04 2019

EXAMPLE

Let F(x) = 1 + x + x^2/(1!*2!) + x^3/(1!*2!*3!) + x^4/(1!*2!*3!*4!) +...+ x^n/G(n) +...

then

F(x)^2 = 1 + 2*x + 4*x^2/(1!*2!) + 14*x^3/(1!*2!*3!) + 122*x^4/(1!*2!*3!*4!) + 3122*x^5/(1!*2!*3!*4!*5!) +...+ a(n)*x^n/G(n) +...

Illustration of initial terms:

a(3) = 1 + 3! + 3! + 1 = 14;

a(4) = 1 + 4! + 4!*3!/2! + 4! + 1 = 122;

a(5) = 1 + 5! + 5!*4!/2! + 5!*4!/2! + 5! + 1 = 3122;

a(6) = 1 + 6! + 6!*5!/2! + 6!*5!*4!/(3!*2!) + 6!*5!/2! + 6! + 1 = 260642; ...

MATHEMATICA

Table[Sum[BarnesG[n+2] / (BarnesG[k+2] * BarnesG[n-k+2]), {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Mar 04 2019 *)

PROG

(PARI) {a(n)=sum(k=0, n, prod(j=0, n, j!)/(prod(j=0, k, j!)*prod(j=0, n-k, j!)))}

(PARI) {a(n)=prod(k=1, n, k!)*polcoeff((sum(m=0, n+1, x^m/prod(k=0, m, k!)+x*O(x^n))^2), n)}

(Sage)

from mpmath import mp

mp.dps = 98; mp.pretty = True

def superbinomial(n, k):

    return mp.superfac(n)/(mp.superfac(k)*mp.superfac(n-k))

def A193520(n): return add(superbinomial(n, k) for k in (0..n))

[int(A193520(n)) for n in (0..14)]  # Peter Luschny, Nov 28 2012

CROSSREFS

Cf. A193521, A000178. Row sums of A009963.

Sequence in context: A238638 A240973 A102449 * A102897 A305856 A001527

Adjacent sequences:  A193517 A193518 A193519 * A193521 A193522 A193523

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 29 2011

STATUS

approved

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Last modified May 14 16:48 EDT 2021. Contains 343898 sequences. (Running on oeis4.)