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A069948 a(n) = 1/exp(1) * Sum_{k>=0} (k+n)!^2 / k!^3. 5
1, 5, 87, 2971, 163121, 12962661, 1395857215, 194634226067, 33990369362241, 7247035915622821, 1848636684656077991, 555005864462114884875, 193458213840943964983537, 77399534126148191747554181, 35196002960227350045891984591, 18037244723394790042393195636291 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Peter Luschny, Mar 27 2011: (Start)

Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) then a(n) = 3! [x^3] taylor(B_{n}(x)), where [x^3] denotes the coefficient of x^3 in the Taylor series for B_{n}(x).

a(n) is column 3 of the square array representation of A090210. (End)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..240

K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)

FORMULA

Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation: a(n)=int(x^n*2*BesselK(0,2*sqrt(x))*hypergeom([],[1,1],x)/exp(1), x=0..infinity), n=0,1... Special values of the hypergeometric function of type 2F2: a(n)=exp(-1)*GAMMA(n+1)^2*hypergeom([n+1, n+1], [1, 1], 1). - Karol A. Penson and G. H. E. Duchamp (gduchamp2(AT)free.fr), Jan 09 2007

Recurrence: (8*n-7)*a(n) = (24*n^3 + 3*n^2 - 26*n + 4)*a(n-1) - (n-1)^2*(24*n^3 - 85*n^2 + 66*n + 13)*a(n-2) + (n-1)^2*(8*n+1)*(n-2)^4*a(n-3). - Vaclav Kotesovec, Jul 30 2013

a(n) ~ n^(2*n+1/3)*exp(n^(1/3) + 3*n^(2/3) - 2*n - 2/3)/sqrt(3) * (1 + 41/(54*n^(1/3)) + 13769/(29160*n^(2/3))). - Vaclav Kotesovec, Jul 30 2013

MAPLE

A069948 := proc(n) exp(-x)*n!^2*hypergeom([n+1, n+1], [1, 1], x);

round(evalf(subs(x=1, %), 99)) end:

seq(A069948(n), n=0..13); # Peter Luschny, Mar 30 2011

# second Maple program:

a:= n-> sum((k+n)!^2/k!^3, k=0..infinity)/exp(1):

seq(a(n), n=0..15);  # Alois P. Heinz, May 17 2018

MATHEMATICA

f[n_] := f[n] = Sum[(k + n)!^3/((k + n)!*(k!^3)*E), {k, 0, Infinity}]; Table[ f[n], {n, 0, 13}] (* or *)

Table[n!^2*HypergeometricPFQ[{n + 1, n + 1}, {1, 1}, 1]/Exp[1], {n, 0, 13}] (* Robert G. Wilson v, Jan 11 2007 *)

PROG

(PARI) {default(realprecision, 200)}; for(n=0, 30, print1(round(exp(-1)*(n!)^2*sum(k=0, 500, binomial(n+k, k)^2/k!)), ", ")) \\ G. C. Greubel, May 17 2018

CROSSREFS

Cf. A000110, A020556, A069223, A090210.

Sequence in context: A182963 A220617 A231702 * A316727 A216088 A305001

Adjacent sequences:  A069945 A069946 A069947 * A069949 A069950 A069951

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, May 02 2002

EXTENSIONS

More terms from Robert G. Wilson v, Jan 11 2007

STATUS

approved

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Last modified February 24 07:45 EST 2020. Contains 332199 sequences. (Running on oeis4.)