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A251669
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E.g.f.: exp(9*x*G(x)^8) / G(x) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.
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10
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1, 8, 191, 8310, 537117, 46444164, 5047987707, 662002733394, 101779688986425, 17959176833948928, 3578033935192224951, 794559576204365478318, 194620831940208238831701, 52129134740350115227721340, 15158273263608217360939225587, 4755712518628181890216523759754
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OFFSET
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0,2
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LINKS
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FORMULA
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Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^8 + 7*G'(x)/G(x).
(2) A(x) = F(x/A(x)^8) where F(x) is the e.g.f. of A251699.
(3) A(x) = Sum_{n>=0} A251699(n)*(x/A(x)^8)^n/n! where A251699(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n.
(4) [x^n/n!] A(x)^(8*n+1) = (7*n+1) * (8*n+1)^(n-1) * 9^n.
a(n) = Sum_{k=0..n} 9^k * n!/k! * binomial(9*n-k-2,n-k) * (8*k-1)/(8*n-1) for n>=0.
Recurrence: 128*(2*n-1)*(4*n-3)*(4*n-1)*(8*n-7)*(8*n-5)*(8*n-3)*(8*n-1)*(33480783*n^8 - 453319173*n^7 + 2697889761*n^6 - 9230277240*n^5 + 19886167926*n^4 - 27672715746*n^3 + 24328423881*n^2 - 12365760717*n + 2776106045)*a(n) = 81*(160137547184727*n^16 - 2968899287488272*n^15 + 25604779347830979*n^14 - 136506824772659775*n^13 + 504285657127489314*n^12 - 1371500076773316825*n^11 + 2847804013092225933*n^10 - 4619534029925962572*n^9 + 5937710241656343834*n^8 - 6090889132598477481*n^7 + 4986522977501530773*n^6 - 3228624422259256476*n^5 + 1615386846720554091*n^4 - 595058403096826425*n^3 + 145565831993332122*n^2 - 17972427186502245*n - 2554359808000)*a(n-1) - 387420489*(33480783*n^8 - 185472909*n^7 + 462117474*n^6 - 687717459*n^5 + 680611896*n^4 - 464268429*n^3 + 210617505*n^2 - 51824070*n - 4480)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 7 * 3^(18*n-3) / 8^(8*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...
such that A(x) = exp(9*x*G(x)^8) / G(x)
where G(x) = 1 + x*G(x)^9 is the g.f. of A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
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MATHEMATICA
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Table[Sum[9^k * n!/k! * Binomial[9*n-k-2, n-k] * (8*k-1)/(8*n-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 07 2014 *)
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PROG
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(PARI) {a(n)=local(G=1); for(i=0, n, G = 1 + x*G^9 +x*O(x^n)); n!*polcoeff(exp(9*x*G^8)/G, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-2, n-k) * (8*k-1)/(8*n-1) )}
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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