A356392
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 3, 17, 99, 769, 6877, 70769, 807321, 10366037, 145721531, 2226927405, 36741898267, 651709348653, 12352436747141, 249152882935829, 5320544034698353, 120008265471779529, 2850195632804141203, 71058458112629765449, 1855470903727083981651
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))^(1/(1-x))))
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a356389(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356389(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356391
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) * d^2 ) /k.
Original entry on oeis.org
1, 5, 35, 206, 1654, 13524, 130668, 1262064, 15027696, 178581600, 2407111200, 33276182400, 514020643200, 8130342124800, 144621487584000, 2537556118272000, 49206063078144000, 982811803276800000, 20991083543732736000, 454612169591580672000, 10763306565511514112000
Offset: 1
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Table[n! * Sum[Sum[(-1)^(k/d + 1)*d^2, {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (-x)^k/(k*(1-x^k)^2))/(1-x)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*log(1+x^k))/(1-x)))
A356393
Expansion of e.g.f. ( Product_{k>0} (1+x^k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 4, 27, 188, 1730, 18234, 220206, 2958416, 44470296, 729675720, 13002636240, 249986061192, 5154030469848, 113360272804128, 2648908519611480, 65477559553098240, 1707034986277780800, 46798324479957887424, 1345365460101611611584
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+x^k)^(1/(1-x))))
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a356390(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356390(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356566
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^x.
Original entry on oeis.org
1, 0, 2, 9, 92, 510, 7074, 68040, 1002224, 12529944, 228706920, 3565888920, 71035245192, 1348127454960, 30270949077264, 661700017709640, 16516072112482560, 408336559236083520, 11204399270843020224, 309489391954850336640, 9258803420755891835520
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^x))
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a078306(n) = sumdiv(n, d, (-1)^(n/d+1)*d^2);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a078306(j-1)/(j-1)*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;
Showing 1-5 of 5 results.