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A090357
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Satisfies A^5 = BINOMIAL(A)^4; also equals A090356^4.
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3
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1, 4, 26, 244, 3131, 52600, 1111940, 28559320, 865622825, 30250881420, 1196941704454, 52860066623036, 2576115583371739, 137274420821505776, 7937914900025008984, 494941882189888642832, 33096552232229291234923
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: A(x)^5 = A(x/(1-x))^4/(1-x)^4.
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)*4^k = A094417(n) = 4*A050353(n) for n >= 1.
BINOMIAL(A(x)) = exp( Sum_{n >= 1} c(n)*x^n/n ) where c(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*(-5)^k = A201365(n) = 5*A050353(n) for n >= 1.
A(x) = B(x)^4 and BINOMIAL(A(x)) = B(x)^5 where B(x) = 1 + x + 5*x^2 + 45*x^3 + 495*x^4 + ... is the o.g.f. for A090356. See also A019538. (End)
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MATHEMATICA
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nmax = 16; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^5 - A[x/(1 - x)]^4/(1 - x)^4 + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
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PROG
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(PARI) {a(n)=local(A); if(n<1, 0, A=1+x+x*O(x^n); for(k=1, n, B=subst(A, x, x/(1-x))/(1-x)+x*O(x^n); A=A-A^5+B^4); polcoeff(A, n, x))}
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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