A354503
Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^(1/k) )^exp(x).
Original entry on oeis.org
1, 1, 3, 14, 67, 424, 3093, 26060, 233917, 2427224, 27565317, 339002146, 4450167269, 63343680802, 964189902141, 15769859929260, 270255218753593, 4913097747513800, 94513145955643993, 1904990351069631390, 40153307898034641361, 893402292594225679438
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k))^exp(x)))
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a354506(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354506(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A354504
Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^k )^exp(x).
Original entry on oeis.org
1, 1, 6, 48, 402, 4375, 54595, 777189, 12284188, 215999025, 4132338673, 85640640877, 1910121348674, 45571124446445, 1157169377895739, 31150000798832647, 885481496002286200, 26498034473000080321, 832407848080194500301, 27378188500890922864153
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^exp(x)))
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a354508(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354508(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A347916
E.g.f.: Product_{k>=1} (1 + x^k)^exp(-x).
Original entry on oeis.org
1, 1, 0, 6, 6, 75, 1025, 1225, 43988, 471345, 5084387, 40870181, 866782774, 8473297261, 165871287465, 3934845305287, 23390789927784, 956832091069057, 21869141108144439, 269518811758178785, 8437830353620298346, 220696789738463945981, 3231280243441039496181, 125072102239522472394691
Offset: 0
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^exp(-x))))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(-x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(-x)*sum(k=1, N, x^k/(k*(1-x^(2*k)))))))
Showing 1-3 of 3 results.