A023882
Expansion of g.f.: 1/Product_{n>0} (1 - n^n * x^n).
Original entry on oeis.org
1, 1, 5, 32, 304, 3537, 52010, 895397, 18016416, 410889848, 10523505770, 298220329546, 9274349837081, 313761671751672, 11474635626789410, 450964042480390679, 18954785687060988578, 848386888530723146912
Offset: 0
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m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-k^k*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
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seq(coeff(series(1/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/(1-k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 19 2015 *)
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m=20; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-k^k*x^k))) \\ G. C. Greubel, Oct 30 2018
A294645
a(n) = Sum_{d|n} d^(n+1).
Original entry on oeis.org
1, 9, 82, 1057, 15626, 282252, 5764802, 134480385, 3486843451, 100048830174, 3138428376722, 107006334784468, 3937376385699290, 155572843119354936, 6568408508343827972, 295150156996346511361, 14063084452067724991010, 708236696816416252145973
Offset: 1
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Table[DivisorSigma[n + 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Oct 07 2020 *)
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{a(n) = sigma(n, n+1)}
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N=66; x='x+O('x^N); Vec(sum(k=1, N, k^(k+1)*x^k/(1-(k*x)^k)))
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N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, 1-(k*x)^k)))) \\ Seiichi Manyama, Jun 02 2019
A294653
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j) in powers of x.
Original entry on oeis.org
1, 1, -1, 1, -1, -1, 1, -1, -4, 0, 1, -1, -16, -23, 0, 1, -1, -64, -713, -229, 1, 1, -1, -256, -19619, -64807, -2761, 0, 1, -1, -1024, -531185, -16757533, -9688425, -42615, 1, 1, -1, -4096, -14347883, -4294435855, -30499541197, -2165979799, -758499, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, ...
-1, -1, -1, -1, -1, ...
-1, -4, -16, -64, -256, ...
0, -23, -713, -19619, -531185, ...
0, -229, -64807, -16757533, -4294435855, ...
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rows = 10;
col[k_] := col[k] = CoefficientList[Product[(1 - j^(k*j)*x^j), {j, 1, rows + 3}] + O[x]^(rows + 3), x];
A[n_, k_] := col[k][[n + 1]];
(* or: *)
A[0, ] = 1; A[n, k_] := A[n, k] = -(1/n)*Sum[DivisorSum[j, #^(1 + k*j) &]*A[n - j, k], {j, 1, n}];
Table[A[n - k, k], {n, 0, rows - 1}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2017 *)
A294947
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(-Sum_{j>0} sigma_k(j)*x^j/j) in powers of x.
Original entry on oeis.org
1, 1, -1, 1, -1, -2, 1, -1, -4, -7, 1, -1, -8, -23, -57, 1, -1, -16, -73, -229, -541, 1, -1, -32, -227, -927, -2761, -7126, 1, -1, -64, -697, -3757, -13969, -42615, -108072, 1, -1, -128, -2123, -15207, -70237, -254580, -758499, -1966034
Offset: 0
Square array begins:
1, 1, 1, 1, 1, ...
-1, -1, -1, -1, -1, ...
-2, -4, -8, -16, -32, ...
-7, -23, -73, -227, -697, ...
-57, -229, -927, -3757, -15207, ...
-541, -2761, -13969, -70237, -351361, ...
A292407
Expansion of Product_{k>=1} ((1 - k^k*x^k)/(1 + k^k*x^k)).
Original entry on oeis.org
1, -2, -6, -40, -386, -4952, -77116, -1406164, -29389570, -692122610, -18136684128, -523599632000, -16516245738140, -565284631133600, -20867650682241704, -826570263691236456, -34971594470460748146, -1574134630989710480092
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(1 - k^k*x^k)/(1 + k^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2017 *)
Showing 1-5 of 5 results.