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
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m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018
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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
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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.
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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
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nmax = 15; Table[SeriesCoefficient[Product[(1 + k^n*x^k), {k, 1, nmax}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 12 2017 *)
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{a(n) = polcoeff(prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}
A292305
a(n) = [x^n] Product_{k>=1} (1 + n^k*x^k).
Original entry on oeis.org
1, 1, 4, 54, 512, 9375, 186624, 4117715, 100663296, 3099363912, 100000000000, 3423740047332, 133741506723840, 5451751918660554, 244464150162276352, 11823135040283203125, 590295810358705651712, 31435129951680797038726, 1809934771463640728469504
Offset: 0
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nmax = 20; Table[SeriesCoefficient[Product[(1+n^k*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Flatten[{1, Table[n^n*PartitionsQ[n], {n, 1, 20}]}]
A318247
a(n) = [x^n] Product_{k>=1} (1 + n!*x^k).
Original entry on oeis.org
1, 1, 2, 42, 600, 28920, 374285520, 128100273840, 131101518683520, 143354704247556480, 173401404266683545849388800, 2538767479410416957720411116800, 105287752487031026606448840363801600, 4510685217145833106538730603088118860800, 288804138719404983322786510403231912442931200
Offset: 0
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nmax = 15; Table[SeriesCoefficient[Product[(1+n!*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Showing 1-4 of 4 results.