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A361304
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Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!.
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2
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1, 2, 18, 124, 930, 7146, 55804, 441312, 3521898, 28307510, 228820086, 1858240956, 15149110912, 123905220292, 1016261712240, 8355494725376, 68842600563918, 568266625104498, 4698576694639306, 38906632384471820, 322596353513983626, 2678048134387075560
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*(1 + x)^4).
(3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(4*n) / n! ) is the g.f. of A361306.
(4) a(n) = (n+1) * A361306(n+1) for n >= 0.
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 18*x^2 + 124*x^3 + 930*x^4 + 7146*x^5 + 55804*x^6 + 441312*x^7 + 3521898*x^8 + 28307510*x^9 + ...
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PROG
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(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); A = sum(m=0, n, Dx(m, x^(2*m)*(1+x +O(x^(n+1)))^(4*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Using series reversion (faster) */
{a(n) = my(A=1); A = deriv( serreverse(x - x^2*(1+x +O(x^(n+3)))^4 )); polcoeff(A, n)}
for(n=0, 25, 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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