A261582
Expansion of Product_{k>=1} 1/(1 + 3*x^k).
Original entry on oeis.org
1, -3, 6, -21, 69, -201, 591, -1785, 5406, -16194, 48426, -145380, 436641, -1309611, 3927399, -11783280, 35354139, -106059387, 318165729, -954506190, 2863556475, -8590643832, 25771817454, -77315531169, 231946940175, -695840583126, 2087520715788, -6262562872614
Offset: 0
-
nmax = 40; CoefficientList[Series[Product[1/(1 + 3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*3^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 + 4/QPochhammer[-3, x])[[3]] (* Vladimir Reshetnikov, Nov 20 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
-
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]
-
{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
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
-
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
-
Table[SeriesCoefficient[Product[1/(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
A266857
Expansion of Product_{k>=1} (1 + 3*x^k)^k.
Original entry on oeis.org
1, 3, 6, 27, 48, 132, 324, 651, 1491, 3078, 6447, 12795, 25839, 50088, 96099, 184491, 343920, 640545, 1173609, 2138403, 3850584, 6882354, 12186336, 21423660, 37421757, 64816608, 111637392, 190976859, 324868530, 549265290, 923904711, 1545406077, 2572326510
Offset: 0
-
nmax=50; CoefficientList[Series[Product[(1+3*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
Showing 1-4 of 4 results.
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