A356394
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^(1/(1-x)).
Original entry on oeis.org
1, 1, 6, 51, 452, 5210, 68514, 1032906, 17352320, 323948376, 6594052680, 145585638000, 3461441121192, 88092914635128, 2388119359650192, 68667743686492440, 2086307088847714560, 66762608893508354880, 2243693428523140377024, 78982154604162553529664
Offset: 0
-
nmax = 20; CoefficientList[Series[Product[(1+x^k)^k, {k, 1, nmax}]^(1/(1-x)), {x, 0, nmax}], x] * Range[0,nmax]! (* Vaclav Kotesovec, Aug 07 2022 *)
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^(1/(1-x))))
-
a356391(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356391(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356389
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) ) /k.
Original entry on oeis.org
1, 2, 10, 34, 218, 1308, 10596, 74688, 793152, 7931520, 94504320, 1054218240, 14662840320, 205279764480, 3427909632000, 50923531008000, 907545606912000, 16335820924416000, 323185344975360000, 6220416698689536000, 140360358705186816000, 3087927891514109952000
Offset: 1
-
Table[n! * Sum[Sum[-(-1)^(k/d), {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
Table[n! * Sum[(2*DivisorSigma[0, 2*k] - 3*DivisorSigma[0, k])/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
-
a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);
-
my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k)/(1-x)))
A356390
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) * d ) /k.
Original entry on oeis.org
1, 3, 17, 74, 514, 3564, 30708, 250704, 2780496, 29982240, 373350240, 4639870080, 67024333440, 988156834560, 16914631507200, 271941778483200, 4999620452198400, 94617104704819200, 1925772463506124800, 39245319872575488000, 902004581585737728000
Offset: 1
-
Table[n! * Sum[Sum[(-1)^(k/d + 1)*d, {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
-
a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/k);
-
my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (-x)^k/(k*(1-x^k)))/(1-x)))
-
my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k))/(1-x)))
A354508
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d^2 )/(k * (n-k)!).
Original entry on oeis.org
1, 5, 32, 168, 1189, 8785, 77384, 646296, 7306737, 79997893, 1005481784, 12518370128, 184109233125, 2671256865121, 47934480000112, 754158322407248, 13813898274148737, 262680987222463269, 5518034466415262320, 107988236156057411096, 2605128008760639636677
Offset: 1
-
a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/(k*(n-k)!));
-
my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, (-x)^k/(k*(1-x^k)^2))))
-
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, k*log(1+x^k))))
Showing 1-4 of 4 results.