0,2

Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:

(1) A'(x)/A(x) = G(x)^6 + 5*G'(x)/G(x).

(2) A(x) = F(x/A(x)^6) where F(x) is the e.g.f. of A251697.

(3) A(x) = Sum_{n>=0} A251697(n)*(x/A(x)^6)^n/n! where A251697(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n .

(4) [x^n/n!] A(x)^(6*n+1) = (5*n+1) * (6*n+1)^(n-1) * 7^n .

a(n) = Sum_{k=0..n} 7^k * n!/k! * binomial(7*n-k-2,n-k) * (6*k-1)/(6*n-1) for n>=0.

Recurrence: 72*(2*n-1)*(3*n-2)*(3*n-1)*(6*n-5)*(6*n-1)*(588245*n^6 - 6117748*n^5 + 26651100*n^4 - 62321728*n^3 + 82554122*n^2 - 58646294*n + 17291583)*a(n) = 7*(69206436005*n^12 - 996572678472*n^11 + 6516703994430*n^10 - 25624338676965*n^9 + 67604945463195*n^8 - 126360374558838*n^7 + 171960790012102*n^6 - 171911061779835*n^5 + 125050872537045*n^4 - 63802357502870*n^3 + 20814954345360*n^2 - 3329274812661*n - 3763584000)*a(n-1) - 823543*(588245*n^6 - 2588278*n^5 + 4886035*n^4 - 5129908*n^3 + 3141733*n^2 - 958104*n - 720)*a(n-2). - Vaclav Kotesovec, Dec 07 2014

a(n) ~ 5 * 7^(7*n-3/2) / 6^(6*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

E.g.f.: A(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...

such that A(x) = exp(7*x*G(x)^6) / G(x)

where G(x) = 1 + x*G(x)^7 is the g.f. of A002296:

G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...

Table[Sum[7^k * n!/k! * Binomial[7*n-k-2, n-k] * (6*k-1)/(6*n-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 07 2014 *)

(PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^7 +x*O(x^n)); n!*polcoeff(exp(7*x*G^6)/G, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n) = sum(k=0, n, 7^k * n!/k! * binomial(7*n-k-2, n-k) * (6*k-1)/(6*n-1) )}

for(n=0, 20, print1(a(n), ", "))

Cf. A251577, A251697, A002295.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251668, A251669, A251670.

Sequence in context: A305134 A083432 A006453 * A127213 A129003 A099138

Adjacent sequences: A251664 A251665 A251666 * A251668 A251669 A251670

Paul D. Hanna, Dec 07 2014

approved