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%I #20 Jul 08 2018 16:30:52
%S 1,0,3,1,10,0,35,4,127,0,462,15,1716,0,6440,57,24310,0,92378,210,
%T 352737,0,1352078,798,5200301,0,20058384,3003,77558760,0,300540195,
%U 11468,1166803440,0,4537567657,43759,17672631900,0,68923265697,168080,269128937220,0,1052049481860,646646,4116715368841,0,16123801841550,2496647,63205303218877,0,247959266493500,9657700,973469712824056,0,3824345300380385,37444162,15033633249846102,0,59132290782430712,145422720,232714176627630544,0,916312070471589206,565730729,3609714217008133585,0,14226520737620288370,2203961430,56093138908332566782,0,221256270138418389602,8597528644,873065282167813104916,0,3446310324346635137703,33578000610,13608507434599516007855,0,53753604366668088230810,131282534380
%N a(n) equals the coefficient of x^n in Sum_{m>=0} (x^m + 1/x^m)^m for n > 0.
%C 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.
%H Paul D. Hanna, <a href="/A304638/b304638.txt">Table of n, a(n) for n = 1..300</a>
%F a(4*n + 2) = 0 for n >= 0.
%F a(n) = [x^n] Sum_{m>=0} x^(m^2) * (1 + 1/x^(2*m))^m, for n > 0.
%e 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 + ...
%e RELATED SERIES.
%e The odd bisection of the g.f. begins:
%e (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) + ...
%e The even bisection of the g.f. begins:
%e (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) + ...
%o (PARI) {a(n) = polcoeff( sum(m=1, n, (x^-m + x^m)^m +x*O(x^n)), n, x)}
%o for(n=1, 80, print1(a(n), ", "))
%Y Cf. A316590, A316591, A316592, A316593, A316594, A316595, A316596 (bisection).
%K nonn
%O 1,3
%A _Paul D. Hanna_, May 15 2018
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