A352004
Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k))^(1/prime(k)!).
Original entry on oeis.org
1, 0, 1, 1, 9, 11, 295, 337, 13041, 45550, 1043211, 3359786, 150500053, 440947300, 23238057921, 145733451241, 5097210717873, 29028404123105, 1710073810205317, 8663532297784519, 574604164708374861, 5108822296820280256, 246335435270285805885
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(isprime(k)/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, isprime(d)*(k-1)!/(d-1)!)*x^k/k!))))
A356460
Expansion of e.g.f. Product_{k>0} B(x^k)^k where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 6, 35, 303, 2772, 32903, 410335, 6051692, 95183187, 1675869175, 31437027030, 644157830077, 13976891765137, 325719071472590, 8007861177420275, 208953947981129027, 5725964099963426924, 165258064179632753563, 4987477844227598529047
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^k)))
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a354863(n) = n!*sumdiv(n, d, n/d/d!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354863(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356524
Expansion of e.g.f. Product_{k>0} 1/(1 - k * x^k)^(1/k!).
Original entry on oeis.org
1, 1, 4, 15, 100, 565, 5946, 46039, 605256, 6646329, 103614490, 1320840631, 27185208876, 401901829069, 9042437722878, 168984439301175, 4257225193170256, 85582303577644465, 2593970612953642386, 57441717948059605927, 1862688382990615542900
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k*x^k)^(1/k!))))
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a354849(n) = (n-1)!*sumdiv(n, d, d^(n/d)/(d-1)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354849(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356542
Expansion of e.g.f. Product_{k>0} 1/(1 - k! * x^k)^(1/k!).
Original entry on oeis.org
1, 1, 4, 18, 132, 900, 11160, 100800, 1809360, 25053840, 608428800, 8610386400, 469291838400, 7110609105600, 404607162960000, 13958116204032000, 821937470818464000, 17420311428103584000, 2860701872247483264000, 60029296274562398784000
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k!*x^k)^(1/k!))))
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a356541(n) = sumdiv(n, d, d*d!^(n/d-1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, a356541(j)*v[i-j+1]/(i-j)!)); v;