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A251696
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a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n.
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9
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1, 5, 324, 44928, 9716112, 2870090496, 1077194894400, 490873123897344, 263285585800098048, 162505400851637010432, 113463916253636561519616, 88423664876285081860177920, 76086820231309990402228260864, 71651521268311905104861664903168, 73298071049899905319337719679434752
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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)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5).
(2) A(x) = F(x*A(x)^5) where F(x) = exp(6*x*G(x)^5)/G(x) is the e.g.f. of A251666.
(3) A(x) = ( Series_Reversion( x*G(x)^5 / exp(30*x*G(x)^5) )/x )^(1/5).
E.g.f.: (-LambertW(-30*x)/(30*x))^(1/5) * (1 + LambertW(-30*x)/30). - Vaclav Kotesovec, Dec 07 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + 5*x + 324*x^2/2! + 44928*x^3/3! + 9716112*x^4/4! + 2870090496*x^5/5! +...
such that A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5),
where G(x) = 1 + x*G(x)^6 is the g.f. A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^5) where
F(x) = 1 + 5*x + 74*x^2/2! + 2028*x^3/3! + 83352*x^4/4! + 4607496*x^5/5! +...
F(x) = exp( 6*x*G(x)^5 ) / G(x) is the e.g.f. of A251666.
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MATHEMATICA
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Table[(4*n + 1)*(5*n + 1)^(n - 2)*6^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)
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PROG
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(PARI) {a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^6 +x*O(x^n));
A = ( serreverse( x*G^5 / exp(30*x*G^5) )/x )^(1/5); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [(4*n + 1)*(5*n + 1)^(n - 2)*6^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017
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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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