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A251587
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a(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807).
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12
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1, 1, 9, 202, 7745, 429192, 31383169, 2862173104, 313456891041, 40120056928000, 5880757402778489, 971776774765633536, 178783183332534538849, 36248462186643418851328, 8031317081954231731640625, 1930762515644331053063077888, 500587184017640118192794723009, 139240118930461640299714951839744
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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)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6.
(2) A(x) = F(x*A(x)) where F(x) = exp(7*x*G(x)^6)/G(x)^6 is the e.g.f. of A251577.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251577.
E.g.f.: -LambertW(-7*x) * (7 + LambertW(-7*x))^6 / (x*7^7). - Vaclav Kotesovec, Dec 07 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 9*x^2/2! + 202*x^3/3! + 7745*x^4/4! + 429192*x^5/5! +...
such that A(x) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6
where G(x) = 1 + x*G(x)^5 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 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(7*x*G(x)^6)/G(x)^6,
F(x) = 1 + x + 7*x^2/2! + 133*x^3/3! + 4501*x^4/4! + 224497*x^5/5! +...
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MATHEMATICA
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Table[7^(n - 5)*(n + 1)^(n - 7)*(1296*n^5 + 9720*n^4 + 30555*n^3 +
50665*n^2 + 44621*n + 16807), {n, 0, 50}] (* G. C. Greubel, Nov 10 2017 *)
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PROG
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(PARI) {a(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(G=1, A=1); for(i=1, n, G=1+x*G^7 +x*O(x^n));
for(i=1, n, A = exp(7*x*A * subst(G^6, x, x*A) ) / subst(G^6, x, x*A) ); n!*polcoeff(A, n)}
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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