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%I #6 Oct 15 2017 04:14:45
%S 1,3,5,7,12,11,26,15,51,19,91,23,155,27,232,62,341,35,592,39,656,344,
%T 870,47,1820,51,1431,1441,1843,59,4758,63,2925,4489,3197,71,11899,75,
%U 4466,11376,7650,83,23052,87,12816,25025,7936,95,57133,99,10706,49131,37220,107,79570,2146,62828,89263,15951,119,228096,123,19500,152146,169033,18864,218253,135,267972,246308,75153,143,724159,147,33227,490146,629034,155,512448,159
%N L.g.f.: Sum_{n>=1} x^((2*n-1)^2) / ( (2*n-1) * (1 - x^(2*n))^(2*n-1) ).
%H Paul D. Hanna, <a href="/A293598/b293598.txt">Table of n, a(n) for n = 1..2050</a>
%e L.g.f.: A(x) = x + 3*x^3/3 + 5*x^5/5 + 7*x^7/7 + 12*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + 51*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + 341*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + 2925*x^65/65 +...
%e which may be written as
%e A(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...
%e The coefficient of x^(2^n+1)/(2^n+1) in A(x) for n>=1 begins:
%e [3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, ..., A293599(n), ...].
%t nmax = 80; Table[(CoefficientList[Series[Sum[x^((2*k - 1)^2)/((2*k - 1)*(1 - x^(2*k))^(2*k - 1)), {k, 1, 2*nmax + 1}], {x, 0, 2*nmax + 1}], x] * Range[0, 2*nmax + 1])[[2*n]], {n, 1, nmax}] (* _Vaclav Kotesovec_, Oct 15 2017 *)
%o (PARI) {a(n) = my(A, Ox = O(x^(2*n+1)));
%o A = sum(m=1, sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );
%o (2*n-1)*polcoeff(A, 2*n-1)}
%o for(n=1, 80, print1(a(n), ", "))
%Y Cf. A293129, A293597, A293599.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Oct 12 2017
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