A320564
Expansion of Product_{k>=1} (1 + x^k/(1 - x)^k)^k.
Original entry on oeis.org
1, 1, 3, 10, 30, 87, 249, 705, 1974, 5471, 15032, 40997, 111079, 299151, 801139, 2134251, 5657895, 14930596, 39232009, 102673794, 267692321, 695440442, 1800582809, 4646964755, 11956293758, 30673060344, 78470890246, 200218512582, 509557661691, 1293664233400, 3276659862518
Offset: 0
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seq(coeff(series(mul((1+x^k/(1-x)^k)^k,k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 15 2018
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nmax = 30; CoefficientList[Series[Product[(1 + x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) (1 - x)^k x^k/(k ((1 - x)^k - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]
A307261
Expansion of Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).
Original entry on oeis.org
1, 1, 4, 13, 42, 130, 397, 1197, 3566, 10517, 30760, 89293, 257397, 737220, 2099215, 5945594, 16756258, 47004829, 131286914, 365203797, 1012031772, 2794446326, 7690009600, 21094325177, 57687762889, 157306741287, 427777384499, 1160250104637, 3139067594584, 8472525405830, 22815639395641
Offset: 0
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a:=series(mul(1/(1-k*x^k/(1-x)^k),k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 03 2019
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nmax = 30; CoefficientList[Series[Product[1/(1 - k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
A307679
Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^(1/k).
Original entry on oeis.org
1, 1, 5, 35, 323, 3679, 49819, 781465, 13923545, 277563617, 6118251461, 147715469131, 3875706370315, 109781717161375, 3338229675519803, 108443658227589329, 3747688533281296049, 137273241169036231105, 5311844045472206624005, 216505267421266611639667, 9270689769095765333645651
Offset: 0
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 323*x^4/4! + 3679*x^5/5! + 49819*x^6/6! + 781465*x^7/7! + 13923545*x^8/8! + ...
log(A(x)) = x + 4*x^2/2 + 11*x^3/3 + 27*x^4/4 + 62*x^5/5 + 137*x^6/6 + 296*x^7/7 + 630*x^8/8 + 1326*x^9/9 + ... + A160399(k)*x^k/k + ...
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nmax = 20; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A307574
Expansion of Product_{k>=1} (1 - (x/(1-x))^k)^k.
Original entry on oeis.org
1, -1, -3, -6, -10, -11, 3, 63, 240, 677, 1622, 3415, 6277, 9485, 8917, -9299, -83683, -309568, -902995, -2315518, -5411355, -11662530, -23117627, -41317787, -62820880, -65358588, 29550902, 449154266, 1783671567, 5453429052, 14668699694, 36273441659
Offset: 0
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m = 31; CoefficientList[Series[Product[(1 - (x/(1-x))^k)^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
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N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(x/(1-x))^k)^k))
A320569
a(n) = [x^n] exp(Sum_{k>=1} sigma_n(k)*x^k/(k*(1 - x)^k)).
Original entry on oeis.org
1, 1, 4, 25, 272, 5028, 173754, 11639691, 1488266409, 375932630887, 190981026883402, 191456188687238845, 388595050299100664773, 1602566853459119962711220, 13153292027392201138778117308, 220500920265786114712328027650814, 7523329040995438987558888118224263531
Offset: 0
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seq(coeff(series(mul((1-x^k/(1-x)^k)^(-k^(n-1)),k=1..n),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 15 2018
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Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n, k] x^k/(k (1 - x)^k), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[1/(1 - x^k/(1 - x)^k)^(k^(n - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Showing 1-5 of 5 results.
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