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A323677
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Coefficients of x^(n*(n+2)) in Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1).
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4
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1, -2, 8, -36, 128, -288, 1166, -16048, 77328, -108780, 220440, -5900816, 44395366, -339891804, 898603106, -5623621248, 2160154604, -22569496436, 930080367404, -11435095847280, 54170773716302, -95118392809132, 264120000910222, -449060398964736, 10522002570379158, -174697886314749868, 1461399560338555304, -10156216740712605688, 35485949488006257078, -90157116358607006060, 722914685477655291098
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OFFSET
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0,2
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COMMENTS
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a(n) = A323557(n*(n+2)) for n >= 0.
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LINKS
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FORMULA
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a(n) = [x^(n*(n+2))] Sum_{k>=0} x^k * (1 + x^k)^k / (1 + x^(k+1))^(k+1).
a(n) = [x^(n*(n+2))] Sum_{k>=0} (-x)^k * (1 - x^k)^k / (1 - x^(k+1))^(k+1).
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EXAMPLE
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Given the g.f. of A323557, G(x) = Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1), i.e.,
G(x) = 1/(1 + x) + x*(1 + x)/(1 + x^2)^2 + x^2*(1 + x^2)^2/(1 + x^3)^3 + x^3*(1 + x^3)^3/(1 + x^4)^4 + x^4*(1 + x^4)^4/(1 + x^5)^5 + x^5*(1 + x^5)^5/(1 + x^6)^6 + x^6*(1 + x^6)^6/(1 + x^7)^7 + x^7*(1 + x^7)^7/(1 + x^8)^8 + ...
and writing G(x) as a power series in x starting as
G(x) = 1 + 3*x^2 - 2*x^3 + 2*x^4 + 9*x^6 - 14*x^7 + 8*x^8 + 12*x^10 - 12*x^11 + 15*x^12 - 52*x^13 + 76*x^14 - 36*x^15 + 2*x^16 + 50*x^18 - 104*x^19 + 79*x^20 - 140*x^21 + 324*x^22 - 276*x^23 + 128*x^24 - 144*x^25 + 118*x^26 - 28*x^27 + 72*x^28 - 336*x^29 + 657*x^30 - 802*x^31 + 1184*x^32 - 1568*x^33 + 1086*x^34 - 288*x^35 + 302*x^36 - 1032*x^37 + 1212*x^38 - 480*x^39 + 142*x^40 + ...
then the coefficients of x^(n*(n+2)) in G(x), for n >= 0, form this sequence.
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PROG
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(PARI) {A323557(n) = my(A=sum(m=0, n, x^m * (1 + x^m +x*O(x^n))^m/(1 + x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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