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A251670
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E.g.f.: exp(10*x*G(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.
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10
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1, 9, 242, 11824, 856824, 82986080, 10097121280, 1481787433920, 254874712419200, 50305519571800960, 11209381628379724800, 2783746998856794752000, 762476362390276346060800, 228363072063685762536960000, 74247696727054926125971251200, 26044746725090717967744412672000
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OFFSET
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0,2
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COMMENTS
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In general, sum_{k=0..n} m^k * n!/k! * binomial(m*n-k-2,n-k) * ((m-1)*k-1)/((m-1)*n-1), m>2, is asymptotic to (m-2) * m^(m*n-3/2) / (m-1)^((m-1)*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014
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LINKS
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FORMULA
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Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^9 + 8*G'(x)/G(x).
(2) A(x) = F(x/A(x)^9) where F(x) is the e.g.f. of A251700.
(3) A(x) = Sum_{n>=0} A251700(n)*(x/A(x)^9)^n/n! where A251700(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n.
(4) [x^n/n!] A(x)^(9*n+1) = (8*n+1) * (9*n+1)^(n-1) * 10^n.
a(n) = Sum_{k=0..n} 10^k * n!/k! * binomial(10*n-k-2,n-k) * (9*k-1)/(9*n-1) for n>=0.
Recurrence: 81*(3*n-2)*(3*n-1)*(9*n-8)*(9*n-7)*(9*n-5)*(9*n-4)*(9*n-2)*(9*n-1)*(100000000*n^9 - 1508750000*n^8 + 10158500000*n^7 - 40108637500*n^6 + 102477510000*n^5 - 175985889125*n^4 + 203494963150*n^3 - 153061617555*n^2 + 68057955478*n - 13624029912)*a(n) = 800*(1250000000000000*n^18 - 25734375000000000*n^17 + 248379687500000000*n^16 - 1494668125000000000*n^15 + 6291920187500000000*n^14 - 19707236445312500000*n^13 + 47696214907031250000*n^12 - 91443867836531250000*n^11 + 141240231848528125000*n^10 - 177729148289358906250*n^9 + 183386452781820390625*n^8 - 155416253373710737500*n^7 + 107706559814898413750*n^6 - 60246014246053412750*n^5 + 26474457002621149925*n^4 - 8675686414409435660*n^3 + 1905677176596950796*n^2 - 212632849946745072*n - 10904042717568)*a(n-1) + 10000000000*(100000000*n^9 - 608750000*n^8 + 1688500000*n^7 - 2844137500*n^6 + 3264185000*n^5 - 2692901625*n^4 + 1611256650*n^3 - 663025355*n^2 + 151278318*n + 4536)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 8 * 10^(10*n-3/2) / 3^(18*n-1) * 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 + 9*x + 242*x^2/2! + 11824*x^3/3! + 856824*x^4/4! + 82986080*x^5/5! +...
such that A(x) = exp(10*x*G(x)^9) / G(x)
where G(x) = 1 + x*G(x)^10 is the g.f. of A059958:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
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MATHEMATICA
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Table[Sum[10^k * n!/k! * Binomial[10*n-k-2, n-k] * (9*k-1)/(9*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^10 +x*O(x^n)); n!*polcoeff(exp(10*x*G^9)/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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