OFFSET
1,3
COMMENTS
The coefficient of 1/x^n in Sum_{m>=0} (x^m + 1/x^m)^m equals a(n) for n > 0, while the constant term in the sum increases without limit.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
a(4*n + 2) = 0 for n >= 0.
a(n) = [x^n] Sum_{m>=0} x^(m^2) * (1 + 1/x^(2*m))^m, for n > 0.
EXAMPLE
G.f.: A(x) = x + 3*x^3 + x^4 + 10*x^5 + 35*x^7 + 4*x^8 + 127*x^9 + 462*x^11 + 15*x^12 + 1716*x^13 + 6440*x^15 + 57*x^16 + 24310*x^17 + 92378*x^19 + 210*x^20 + 352737*x^21 + 1352078*x^23 + 798*x^24 + 5200301*x^25 + ...
RELATED SERIES.
The odd bisection of the g.f. begins:
(A(x) - A(-x))/2 = x + 3*x^3 + 10*x^5 + 35*x^7 + 127*x^9 + 462*x^11 + 1716*x^13 + 6440*x^15 + 24310*x^17 + 92378*x^19 + 352737*x^21 + 1352078*x^23 + 5200301*x^25 + 20058384*x^27 + 77558760*x^29 + 300540195*x^31 + 1166803440*x^33 + 4537567657*x^35 + 17672631900*x^37 + 68923265697*x^39 + 269128937220*x^41 + 1052049481860*x^43 + 4116715368841*x^45 + 16123801841550*x^47 + 63205303218877*x^49 + ... + A316596(n)*x^(2*n-1) + ...
The even bisection of the g.f. begins:
(A(x) + A(-x))/2 = x^4 + 4*x^8 + 15*x^12 + 57*x^16 + 210*x^20 + 798*x^24 + 3003*x^28 + 11468*x^32 + 43759*x^36 + 168080*x^40 + 646646*x^44 + 2496647*x^48 + 9657700*x^52 + 37444162*x^56 + 145422720*x^60 + 565730729*x^64 + 2203961430*x^68 + 8597528644*x^72 + 33578000610*x^76 + 131282534380*x^80 + ... + A316592(n)*x^(4*n) + ...
PROG
(PARI) {a(n) = polcoeff( sum(m=1, n, (x^-m + x^m)^m +x*O(x^n)), n, x)}
for(n=1, 80, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 15 2018
STATUS
approved