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A251587 a(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807). 12
1, 1, 9, 202, 7745, 429192, 31383169, 2862173104, 313456891041, 40120056928000, 5880757402778489, 971776774765633536, 178783183332534538849, 36248462186643418851328, 8031317081954231731640625, 1930762515644331053063077888, 500587184017640118192794723009, 139240118930461640299714951839744 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..300

FORMULA

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

EXAMPLE

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! +...

is the e.g.f. of A251577.

MATHEMATICA

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 *)

PROG

(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), ", "))

CROSSREFS

Cf. A251577, A002296.

Cf. Variants: A251583, A251584, A251585, A251586, A251588, A251589, A251590.

Sequence in context: A012133 A012038 A012108 * A103914 A211039 A307735

Adjacent sequences:  A251584 A251585 A251586 * A251588 A251589 A251590

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 06 2014

STATUS

approved

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Last modified January 16 17:49 EST 2022. Contains 350376 sequences. (Running on oeis4.)