A261584
Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - 2*x^k).
Original entry on oeis.org
1, 4, 12, 36, 92, 228, 540, 1236, 2748, 6004, 12876, 27252, 57036, 118308, 243564, 498564, 1015484, 2060484, 4167804, 8409588, 16934748, 34049940, 68378220, 137185428, 275026476, 551052676, 1103618508, 2209525092, 4422484764, 8850120420, 17707920924
Offset: 0
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nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1)), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 - QPochhammer[-2, x]/(3 QPochhammer[2, x]))[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)
A261562
Expansion of Product_{k>=1} (1 + 2*x^k)^k.
Original entry on oeis.org
1, 2, 4, 14, 24, 58, 124, 238, 480, 922, 1764, 3238, 6008, 10794, 19292, 34166, 59504, 103042, 176452, 299958, 505240, 845570, 1403324, 2315118, 3794640, 6180370, 10009540, 16121374, 25829512, 41171690, 65320956, 103140062, 162149488, 253823178, 395698276
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(2^j*binomial(i, j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..40); # Alois P. Heinz, Sep 21 2018
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nmax = 50; CoefficientList[Series[Product[(1 + 2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 50; s = 1+2*x; Do[s*=Sum[Binomial[k, j]*2^j*x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Jan 08 2016 *)
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{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-2)^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015
A261561
Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.
Original entry on oeis.org
1, 2, 8, 22, 64, 162, 424, 1022, 2480, 5770, 13336, 30046, 67184, 147554, 321592, 692278, 1479568, 3133474, 6596008, 13788606, 28679264, 59335530, 122256456, 250875550, 513116864, 1046190786, 2127557592, 4316282006, 8739096992, 17661731138, 35639764536
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(2^j*binomial(i+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..40); # Alois P. Heinz, Sep 21 2018
-
nmax = 50; CoefficientList[Series[Product[(1/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]
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{a(n) = polcoeff( exp( sum(m=1,n,x^m/m * sumdiv(m,d,2^d*m^2/d^2) ) +x*O(x^n)),n)}
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Sep 30 2015
A300412
a(n) = [x^n] Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k.
Original entry on oeis.org
1, 2, 16, 144, 1376, 15800, 210816, 3333372, 61688448, 1318588146, 32004369200, 869282342632, 26099925704928, 857736429098848, 30605729417479104, 1177841009504482200, 48614265201514729984, 2141639401723095243324, 100282931820560447963568, 4973060138191518242569120
Offset: 0
The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (2), 6, 16, 38, 88, ...
n = 2: 1, 4, (16), 60, 192, 596, ...
n = 3: 1, 6, 30, (144), 582, 2280, ...
n = 4: 1, 8, 48, 280, (1376), 6568, ...
n = 5: 1, 10, 70, 480, 2790, (15800), ...
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Table[SeriesCoefficient[Product[((1 + n x^k)/(1 - n x^k))^k, {k, 1, n}], {x, 0, n}], {n, 0, 19}]
Showing 1-4 of 4 results.