A340103
a(n) = [x^n] Product_{k>=1} (1 + n^(k-1)*x^k).
Original entry on oeis.org
1, 1, 2, 12, 80, 875, 10584, 170471, 2949120, 63772920, 1441000000, 38818444632, 1089573617664, 35185728919614, 1175820172477440, 44425722744140625, 1722925924631969792, 74364737115532234518, 3291298649632850485248, 159785357022861166517580, 7932051456000000000000000
Offset: 0
-
Table[SeriesCoefficient[Product[(1 + n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] n^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 20}]
Join[{1}, Table[SeriesCoefficient[n*QPochhammer[-1/n, n*x]/(n+1), {x, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 09 2021 *)
A265949
Expansion of Product_{k>=1} (1 + k^k*x^k).
Original entry on oeis.org
1, 1, 4, 31, 283, 3489, 50913, 890635, 17891170, 409850236, 10494427982, 297780829216, 9261266862273, 313453533534739, 11464487066049791, 450644378868285130, 18942868694407904729, 847930346323808122469, 40266107916200371331007, 2021842180288047801103956
Offset: 0
-
m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018
-
seq(coeff(series(mul((1+k^k*x^k),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018
-
nmax=20; CoefficientList[Series[Product[(1+k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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m=30; x='x+O('x^m); Vec(prod(k=1, m, (1+k^k*x^k))) \\ G. C. Greubel, Oct 31 2018
A292190
Sum of n-th powers of products of terms in all partitions of n into distinct parts.
Original entry on oeis.org
1, 1, 4, 35, 337, 11925, 371081, 49032439, 3545396034, 3416952655320, 749189363202730, 598250899004413536, 2383502427069445040595, 1729793152213690218766715, 131751643363739706679145099315, 271212858254426215135033141804302
Offset: 0
5 = 4 + 1 = 3 + 2. So a(5) = 5^5 + (4*1)^5 + (3*2)^5 = 11925.
-
b:= proc(n, i, k) option remember; (m->
`if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
end:
a:= n-> b(n$3):
seq(a(n), n=0..20); # Alois P. Heinz, Sep 11 2017
-
nmax = 15; Table[SeriesCoefficient[Product[(1 + k^n*x^k), {k, 1, nmax}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 12 2017 *)
-
{a(n) = polcoeff(prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}
A292306
a(n) = [x^n] Product_{k>=1} (1 + n^n*x^k).
Original entry on oeis.org
1, 1, 4, 756, 65792, 19534375, 101564310279744, 558547898753326097, 9444733810164237336576, 174449211609498720646587480, 10000000004000000000400000000010000000000, 6626407607852766876000106671521201448502431912
Offset: 0
-
nmax = 14; Table[SeriesCoefficient[Product[(1+n^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
A344094
a(n) = [x^n] Product_{k>=1} (1 + n^(k+1)*x^k).
Original entry on oeis.org
1, 1, 8, 324, 5120, 171875, 13716864, 409300871, 20535312384, 976299632280, 144100000000000, 6251749326428232, 484144254340300800, 31585633366079696358, 2452531026468909711360, 483966896057281494140625, 31314307295813796764844032, 3176091371161687418319418614
Offset: 0
-
Table[SeriesCoefficient[Product[(1+n^(k+1)*x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[SeriesCoefficient[QPochhammer[-n, n*x]/(n+1), {x, 0, n}], {n, 1, 20}]]
Showing 1-5 of 5 results.
