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A251585
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a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125).
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9
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1, 1, 7, 116, 3229, 129000, 6776875, 443200000, 34766465625, 3185000000000, 333992093359375, 39470976000000000, 5192072114658203125, 752537122540000000000, 119176291179656982421875, 20476256583680000000000000, 3793880513498167242431640625, 754086862404270000000000000000
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OFFSET
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0,3
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LINKS
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FORMULA
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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) = exp( 5*x*A(x) * G(x*A(x))^4 ) / G(x*A(x))^4.
(2) A(x) = F(x*A(x)) where F(x) = exp(5*x*G(x)^4)/G(x)^4 is the e.g.f. of A251575.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251575.
E.g.f.: -LambertW(-5*x) * (5 + LambertW(-5*x))^4 / (x*5^5). - Vaclav Kotesovec, Dec 07 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 7*x^2/2! + 116*x^3/3! + 3229*x^4/4! + 129000*x^5/5! + ...
such that A(x) = exp( 5*x*A(x) * G(x*A(x))^4 ) / G(x*A(x))^4
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 + ...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(5*x*G(x)^4)/G(x)^4,
F(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1505*x^4/4! + 51505*x^5/5! + ...
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MATHEMATICA
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Table[5^(n - 3)*(n + 1)^(n - 5)*(16*n^3 + 87*n^2 + 172*n + 125), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)
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PROG
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(PARI) {a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(G=1, A=1); for(i=1, n, G=1+x*G^5 +x*O(x^n));
for(i=1, n, A = exp(5*x*A * subst(G^4, x, x*A) ) / subst(G^4, x, x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [5^(n - 3)*(n + 1)^(n - 5)*(16*n^3 + 87*n^2 + 172*n + 125): n in [0..50]]; // G. C. Greubel, Nov 13 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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