A209903
E.g.f.: Product_{n>=1} B(x^n) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 4, 17, 111, 752, 6893, 64171, 733540, 8751579, 119847295, 1716294780, 27583937857, 460405876777, 8428298492136, 160944930254405, 3309210789416387, 70814345769448444, 1617322515279759301, 38322855872232745163, 960820910852189283072
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 111*x^4/4! + 752*x^5/5! +...
Let B(x) = exp(exp(x)-1) be the e.g.f. of Bell numbers:
B(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...
then the e.g.f. of this sequence equals the infinite product:
A(x) = B(x)*B(x^2)*B(x^3)*B(x^4)*B(x^5)*B(x^6)...
The logarithm of the e.g.f. A(x) begins:
log(A(x)) = x + 3*x^2/2! + 7*x^3/3! + 37*x^4/4! + 121*x^5/5! + 1201*x^6/6! +...+ A057625(n)*x^n/n! +...
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{a(n)=local(Bell=exp(exp(x+x*O(x^n))-1));n!*polcoeff(prod(m=1,n,subst(Bell,x,x^m+x*O(x^n))),n)}
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{a(n)=n!*polcoeff(exp(sum(m=1,n,x^m/m!/(1-x^m+x*O(x^n)))),n)}
for(n=0,25,print1(a(n),", "))
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a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/d!)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Jul 02 2021
A345762
E.g.f.: Product_{k>=1} (1 - x^k)^(1/k!).
Original entry on oeis.org
1, -1, -1, 2, 0, 29, -135, 727, -1967, -6074, 94510, 1548051, -41361089, 408842095, 213929807, -41951737904, 130060640466, 10569226878107, -229371598130229, 3327344803563111, -31418096993670379, -383829978086171112, 17799865170898698140, 220582224147105677385
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1-x^k)^(1/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, (exp(x^k)-1)/k))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d-1)!)*x^k/k))))
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a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, sumdiv(k, d, 1/(d-1)!)*a(n-k)/(n-k)!));
A346057
Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k)).
Original entry on oeis.org
1, -1, -1, 2, 3, 14, -55, 62, -637, 338, -3861, 335312, -4499803, 43490108, -246353731, 2189950310, -47336985225, 1224524919590, -21516426400621, 346681988108648, -4499477383730851, 69294602646065900, -1418045089870455795, 45246859024830444566
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k/k)))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, 1-exp(x^k/k)))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)^d))*x^k))))
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a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)^d))*a(n-k)/(n-k)!));
A346058
Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k!)).
Original entry on oeis.org
1, -1, -1, 3, 4, 2, -69, -185, 596, 1482, 22051, -8341, -450570, -1503596, -23829233, 144974757, 150086353, 4859956733, 51013196234, -504522222442, 2572161050316, -58533039862692, 69278113622988, 342581575176372, -25348876024693055, 661312712021911319
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k/k!)))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, 1-exp(x^k/k!)))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!^d))*x^k))))
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a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!^d))*a(n-k)/(n-k)!));
A346039
Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k))^(1/k!).
Original entry on oeis.org
1, -1, -1, 3, 1, 17, -119, 165, 1191, -21989, 169527, -317837, -7182779, 54452161, 292654649, -4320853051, -46883217705, 728176373539, 9943868087879, -166076498591597, -2748733072385043, 65290726021558089, 151614363753006601, -11661992771499644571
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k))^(1/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (1-exp(x^k))/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!))*x^k))))
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a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!))*a(n-k)/(n-k)!));
Showing 1-5 of 5 results.