OFFSET
0,2
COMMENTS
Sum_{k = 0..n} A094816(n,k)*x^k give A000522(n), A001339(n), A082030(n), A095000(n) for x = 1, 2, 3, 4 respectively.
From Peter Bala, Jul 10 2008: (Start)
a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.
Recurrence relation: a(0) = 1, a(1) = 6, a(n) = (n+5)*a(n-1) - (n-1)*a(n-2) for n >= 2. Let p_4(n) = n^4+2*n^3+5*n^2+1 = n^(4)-4*n^(3)+6*n^(2)-4*n^(1)+1, where n^(k) denotes the rising factorial n*(n+1)*...*(n+k-1). The polynomial p_4(n) is an example of a Poisson-Charlier polynomial c_k(x;a) at k = 4, x = -n and a = -1.
The sequence b(n) := n!*p_4(n+1) = A001688(n) satisfies the same recurrence as a(n) but with the initial conditions b(0) = 9, b(1) = 53. This leads to the finite continued fraction expansion expansion a(n)/b(n) = 1/(9-1/(6-1/(7-2/(8-...-(n-1)/(n+5))))).
Lim n -> infinity a(n)/b(n) = e/24 = 1/(9-1/(6-1/(7-2/(8-...-n/((n+6)-...))))).
a(n) = b(n) * sum {k = 0..n} 1/(k!*p_4(k)*p_4(k+1)) - since the rhs satisfies the above recurrence with the same initial conditions. Hence e = 24 * sum {k = 0..inf} 1/(k!*p_4(k)p_4(k+1)).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Poisson-Charlier polynomial
FORMULA
a(n) = Sum_{k = 0..n} A094816(n, k)*5^k.
a(n) = Sum_{k=0..n} binomial(n, k)*(k+4)!/4!.
G.f.: 1/Q(0), where Q(k) = 1 - x - x*(k+5)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
a(n) ~ n! *exp(1)*n^4/24. - Vaclav Kotesovec, Jun 21 2013
a(n) = 2F0(5,-n;;-1). - Benedict W. J. Irwin, May 27 2016
First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) + 1 with a(0) = 1, where P(n) = n^4 + 6*n^3 + 17*n^2 + 20*n + 9 = A094793(n). - Peter Bala, Jul 26 2021
MATHEMATICA
CoefficientList[Series[Exp[x]/(1-x)^5, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 21 2013 *)
Table[HypergeometricPFQ[{5, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*(k+4)!/4! ); \\ Joerg Arndt, Apr 22 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Deléham, Jun 20 2004
STATUS
approved