A346055
Expansion of e.g.f. Product_{k>=1} B(x^k/k) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 3, 10, 47, 246, 1617, 11586, 97463, 891610, 9189623, 102024396, 1250714445, 16351489116, 232261545869, 3499469551402, 56582677946675, 964734301550142, 17509882651329087, 333381717125596692, 6710286637806825557, 141167551783524139468
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k/k)-1))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, exp(x^k/k)-1))))
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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)!));
A346056
Expansion of e.g.f. Product_{k>=1} B(x^k/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 3, 9, 38, 168, 915, 5225, 34228, 236622, 1805297, 14498751, 125907798, 1146476984, 11129874215, 112934907867, 1209762361679, 13499714095281, 157931096158594, 1918777335806274, 24309294470496502, 318987321135326838, 4346474397776153974
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k/k!)-1))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, exp(x^k/k!)-1))))
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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)!));
A352607
Triangle read by rows. T(n, k) = Bell(k)*Sum_{j=0..k}(-1)^(k+j)*binomial(n, n-k+j)*Stirling2(n-k+j, j) for n >= 0 and 0 <= k <= floor(n/2).
Original entry on oeis.org
1, 0, 0, 1, 0, 1, 0, 1, 6, 0, 1, 20, 0, 1, 50, 75, 0, 1, 112, 525, 0, 1, 238, 2450, 1575, 0, 1, 492, 9590, 18900, 0, 1, 1002, 34125, 141750, 49140, 0, 1, 2024, 114675, 854700, 900900, 0, 1, 4070, 371580, 4544925, 9909900, 2110185
Offset: 0
Triangle starts:
[0] 1;
[1] 0;
[2] 0, 1;
[3] 0, 1;
[4] 0, 1, 6;
[5] 0, 1, 20;
[6] 0, 1, 50, 75;
[7] 0, 1, 112, 525;
[8] 0, 1, 238, 2450, 1575;
[9] 0, 1, 492, 9590, 18900;
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A352607 := (n, k) -> combinat:-bell(k)*add((-1)^(k+j)*binomial(n, n-k+j)* Stirling2(n-k+j, j), j = 0..k):
seq(seq(A352607(n, k), k = 0..n/2), n = 0..12);
# Second program:
egf := k -> combinat[bell](k)*(exp(x) - 1 - x)^k/k!:
A352607 := (n, k) -> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A352607(n, k), k = 0..n/2)), n=0..12);
# Recurrence:
A352607 := proc(n, k) option remember;
if k > n/2 then 0 elif k = 0 then k^n else k*A352607(n-1, k) +
combinat[bell](k)/combinat[bell](k-1)*(n-1)*A352607(n-2, k-1) fi end:
seq(print(seq(A352607(n, k), k=0..n/2)), n=0..12); # Mélika Tebni, Mar 24 2022
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T[n_, k_] := BellB[k]*Sum[(-1)^(k+j)*Binomial[n, n-k+j]*StirlingS2[n-k+j, j], {j, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Oct 21 2023 *)
A352608
a(n) = Bell(n)*n!!.
Original entry on oeis.org
1, 1, 4, 15, 120, 780, 9744, 92085, 1589760, 19983915, 445344000, 7053735150, 194162549760, 3735730993995, 123152970608640, 2803291544678625, 108175188829962240, 2855475766145702700, 126726160086516695040, 3818865908630429932275, 192201344414226343526400
Offset: 0
Showing 1-4 of 4 results.