A354610
Expansion of e.g.f. exp(f(x) - 1) where f(x) = (1 - x)^x = e.g.f. for A007114.
Original entry on oeis.org
1, 0, -2, -3, 16, 90, -84, -2940, -8672, 95256, 956160, -811800, -75724296, -419150160, 4406562720, 78306555600, 89704074240, -9655388184960, -97621097227200, 657339885653760, 23680733504400000, 119677890314505600, -3528587069869276800, -64401874868363598720
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1-x)^x-1)))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j!*sum(k=0, j\2, (-1)^(j-k)*stirling(j-k, k, 1)/(j-k)!)*binomial(i-1, j-1)*v[i-j+1])); v;
A318616
a(n) = n! * [x^n] (1 - x)^(n*x).
Original entry on oeis.org
1, 0, -4, -9, 160, 1350, -14904, -335160, 1796096, 125615448, 204300000, -64591072920, -735003528192, 41673388751280, 1113912529707264, -30043364514345000, -1703374149711298560, 17822402097051182400, 2856178489894627203072, 12394040043610922716800, -5255899207995216384000000
Offset: 0
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Table[n! SeriesCoefficient[(1 - x)^(n x), {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[n! Sum[(-1)^k n^(n - k) StirlingS1[k, n - k]/k!, {k, n}], {n, 20}]]
A354611
Expansion of e.g.f. 1/(2 - (1 - x)^x).
Original entry on oeis.org
1, 0, -2, -3, 28, 150, -714, -10920, 13392, 1129464, 3694680, -150143400, -1515256104, 22631946480, 525582087408, -2756199995640, -192774443051520, -525316900812480, 75951597634314048, 926307802605928320, -30597152030347651200, -833744424171043728000
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(2-(1-x)^x)))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j!*sum(k=0, j\2, (-1)^(j-k)*stirling(j-k, k, 1)/(j-k)!)*binomial(i, j)*v[i-j+1])); v;
Showing 1-3 of 3 results.