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A090358 Satisfies A^6 = BINOMIAL(A^5). 9
1, 1, 6, 66, 1071, 23151, 627236, 20452976, 779947641, 34050858041, 1674497370602, 91575747294582, 5512402585832847, 362148111801511407, 25783279860096503952, 1977349647140061768364, 162508269041154881377519 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

In general, if A^n = BINOMIAL(A^(n-1)), then for all integer m>0 there exists an integer sequence B such that B^d = BINOMIAL(A^m) where d=gcd(m+1,n). Also, coefficients of A(k*x)^n = k-th binomial transform of coefficients in A(k*x)^(n-1) for all k>0.

In general, if g.f. satisfies A(x)^(k+1) = A(x/(1-x))^k / (1-x), k>0, then a(n) ~ (n-1)! / (k*(k+1) * (log((k+1)/k))^(n+1)). - Vaclav Kotesovec, Nov 19 2014

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..280

FORMULA

G.f. satisfies: A(x)^6 = A(x/(1-x))^5/(1-x).

a(n) ~ (n-1)! / (30 * (log(6/5))^(n+1)). - Vaclav Kotesovec, Nov 19 2014

O.g.f. A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = Sum_{k = 1..n} k!*Stirling2(n,k)*5^(k-1) = 1/5*A094418(n) for n >= 1. - Peter Bala, May 26 2015

EXAMPLE

A^6 = BINOMIAL(A090362), since A090362=A^5. Also,

BINOMIAL(A) = A090359^2 since 2=gcd(1+1,6),

BINOMIAL(A^2) = A090360^3 since 3=gcd(2+1,6) and

BINOMIAL(A^3) = A090361^2 since 2=gcd(3+1,6).

PROG

(PARI) {a(n)=local(A); if(n<1, 0, A=1+x+x*O(x^n); for(k=1, n, B=subst(A^5, x, x/(1-x))/(1-x)+x*O(x^n); A=A-A^6+B); polcoeff(A, n, x))}

CROSSREFS

Cf. A084784, A090351, A090353, A090356, A090359, A090360, A090361, A090362, A094418.

Sequence in context: A128319 A174496 A008548 * A264407 A112942 A113390

Adjacent sequences:  A090355 A090356 A090357 * A090359 A090360 A090361

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Nov 26 2003

STATUS

approved

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Last modified February 22 15:52 EST 2019. Contains 320399 sequences. (Running on oeis4.)