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
A261563
Expansion of Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^k.
Original entry on oeis.org
1, 4, 16, 60, 192, 596, 1744, 4892, 13248, 34868, 89296, 223660, 548928, 1323060, 3137520, 7332332, 16907584, 38517444, 86777328, 193523404, 427562816, 936555044, 2035286576, 4390850268, 9409096576, 20037827876, 42429318480, 89369282460, 187325508288
Offset: 0
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nmax = 50; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1))^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 - (-2)^d) * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Sep 30 2015
A261565
Expansion of Product_{k>=1} (1/(1 - 3*x^k))^k.
Original entry on oeis.org
1, 3, 15, 54, 201, 672, 2268, 7266, 23208, 72414, 224652, 688929, 2103975, 6386907, 19337091, 58367817, 175905741, 529331190, 1591515297, 4781575074, 14359673454, 43108645230, 129387584991, 388283978589, 1165099808574, 3495782937135, 10488322595625
Offset: 0
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nmax = 40; CoefficientList[Series[Product[(1/(1 - 3*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[3^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, 3^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015
A298985
a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k.
Original entry on oeis.org
1, 1, 8, 54, 496, 5400, 73728, 1204322, 23167808, 512093178, 12781430600, 355128859129, 10863077554224, 362572265689777, 13107541496092960, 510105773344747725, 21258690342206888192, 944467894258279964254, 44555341678790400325512, 2224158766859058600584834, 117123916650423288611260400
Offset: 0
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(i+j-1, j)*b(n-i*j, i-1, k)*k^j, j=0..n/i)))
end:
a:= n-> b(n$3):
seq(a(n), n=0..30); # Alois P. Heinz, Sep 23 2018
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Table[SeriesCoefficient[Product[1/(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Showing 1-4 of 4 results.
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