A251665 E.g.f.: exp(5*x*G(x)^4) / G(x) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

1, 4, 47, 1034, 34349, 1540480, 87311275, 5991370390, 483100288985, 44778459212540, 4691799973171175, 548418557098305250, 70754785462138421125, 9987462340422594014200, 1531136319790275407365475, 253347224928445454055920750, 45001449932636667231257800625, 8541130421294458307989700672500

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:

(1) A'(x)/A(x) = G(x)^4 + 3*G'(x)/G(x).

(2) A(x) = F(x/A(x)^4) where F(x) is the e.g.f. of A251695.

(3) A(x) = Sum_{n>=0} A251695(n)*(x/A(x)^4)^n/n! where A251695(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n .

(4) [x^n/n!] A(x)^(4*n+1) = (3*n+1) * (4*n+1)^(n-1) * 5^n.

a(n) = Sum_{k=0..n} 5^k * n!/k! * binomial(5*n-k-2,n-k) * (4*k-1)/(4*n-1) for n>=0.

Recurrence: 8*(2*n-1)*(4*n-3)*(4*n-1)*(1875*n^4 - 13375*n^3 + 35700*n^2 - 41905*n + 17681)*a(n) = 5*(1171875*n^8 - 11875000*n^7 + 51765625*n^6 - 126596875*n^5 + 189126875*n^4 - 174442875*n^3 + 93137550*n^2 - 22362645*n - 233856)*a(n-1) - 3125*(1875*n^4 - 5875*n^3 + 6825*n^2 - 3130*n - 24)*a(n-2). - Vaclav Kotesovec, Dec 07 2014

a(n) ~ 3 * 5^(5*n-3/2) / 2^(8*n-1) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

Example

E.g.f.: A(x) = 1 + 4*x + 47*x^2/2! + 1034*x^3/3! + 34349*x^4/4! + 1540480*x^5/5! +...
such that A(x) = exp(5*x*G(x)^4) / G(x)
where G(x) = 1 + x*G(x)^5 is the g.f. of A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...

Mathematica

Table[Sum[5^k * n!/k! * Binomial[5*n-k-2, n-k] * (4*k-1)/(4*n-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 07 2014 *)

Prog

(PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^5 +x*O(x^n)); n!*polcoeff(exp(5*x*G^4)/G, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, 5^k * n!/k! * binomial(5*n-k-2, n-k) * (4*k-1)/(4*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A251575, A251695, A002294.

Cf. Variants: A243953, A251663, A251664, A251666, A251667, A251668, A251669, A251670.

Sequence in context: A326434 A319833 A006438 * A210828 A361560 A333247

Adjacent sequences: A251662 A251663 A251664 * A251666 A251667 A251668

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2014

Status

approved