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A341375
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: [Sum_{n>=0} x^n/(1 - x^(n+1))]^4 = Sum_{n>=0} a(n)*x^n/(1 - x^(n+1))^4.
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2
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1, 4, 22, 56, 181, 256, 730, 968, 1859, 2368, 4810, 4680, 9674, 9944, 15024, 17580, 29693, 25792, 47226, 42476, 62505, 66388, 104586, 83700, 141472, 132768, 181046, 173964, 273121, 202124, 359578, 307860, 410700, 399924, 551552, 436980, 743931, 629304, 816002
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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 satisfies:
(1) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^4 = Sum_{n>=0} a(n) * x^n / (1 - x^(n+1))^4.
(2) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^4 = Sum_{n>=0} (n+1)*(n+2)*(n+3)/6 * x^n * A( x^(n+1) ).
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EXAMPLE
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A(x) = 1 + 4*x + 22*x^2 + 56*x^3 + 181*x^4 + 256*x^5 + 730*x^6 + 968*x^7 + 1859*x^8 + 2368*x^9 + 4810*x^10 + 4680*x^11 + 9674*x^12 + ...
such that
D(x)^4 = 1/(1-x)^4 + 4*x/(1-x^2)^4 + 22*x^2/(1-x^3)^4 + 56*x^3/(1-x^4)^4 + 181*x^4/(1-x^5)^4 + 256*x^5/(1-x^6)^4 + ... + a(n)*x^n/(1-x^(n+1))^4 + ...
and
D(x)^4 = A(x) + 4*x*A(x^2) + 10*x^2*A(x^3) + 20*x^3*A(x^4) + 35*x^4*A(x^5) + 56*x^5*A(x^6) + 84*x^6*A(x^7) + ... + (n+1)*(n+2)*(n+3)/6*x^n*A(x^(n+1)) + ...
where
D(x)^4 = 1 + 8*x + 32*x^2 + 92*x^3 + 216*x^4 + 440*x^5 + 814*x^6 + 1392*x^7 + 2244*x^8 + 3452*x^9 + 5096*x^10 + 7292*x^11 + 10129*x^12 + ...
D(x) = 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 4*x^9 + 2*x^10 + 6*x^11 + 2*x^12 + ... + A000005(n+1)*x^n + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A, x^n/(1 - x^(n+1) +x*O(x^#A)) )^4 - sum(n=0, #A-1, A[n+1]*x^n/(1 - x^(n+1) + x*O(x^#A))^4 ), #A-1) ); A[n+1]}
for(n=0, 100, 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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