A350256
Triangle read by rows. T(n, k) = BellPolynomial(n, k).
Original entry on oeis.org
1, 0, 1, 0, 2, 6, 0, 5, 22, 57, 0, 15, 94, 309, 756, 0, 52, 454, 1866, 5428, 12880, 0, 203, 2430, 12351, 42356, 115155, 268098, 0, 877, 14214, 88563, 355636, 1101705, 2869242, 6593839, 0, 4140, 89918, 681870, 3188340, 11202680, 32510850, 82187658, 187104200
Offset: 0
Triangle begins:
[0] 1
[1] 0, 1
[2] 0, 2, 6
[3] 0, 5, 22, 57
[4] 0, 15, 94, 309, 756
[5] 0, 52, 454, 1866, 5428, 12880
[6] 0, 203, 2430, 12351, 42356, 115155, 268098
[7] 0, 877, 14214, 88563, 355636, 1101705, 2869242, 6593839
[8] 0, 4140, 89918, 681870, 3188340, 11202680, 32510850, 82187658, 187104200
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A350256 := (n, k) -> ifelse(n = 0, 1, BellB(n, k)):
seq(seq(A350256(n, k), k = 0..n), n = 0..8);
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T[n_, k_] := BellB[n, k]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
A299824
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)!.
Original entry on oeis.org
2, 22, 309, 5428, 115155, 2869242, 82187658, 2661876168, 96202473183, 3838516103310, 167606767714397, 7949901069639228, 407048805012563038, 22376916254447538882, 1314573505901491675965, 82188946843192555474704, 5448870914168179374456623, 381819805747937892412056342
Offset: 1
a(4) = (1/e^4)*Sum_{j >= 1} j^4 * 4^j / (j-1)! = 5428.
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a(n) = round(exp(-n)*suminf(j = 1, (j^n)*(n^j)/(j-1)!)); \\ Michel Marcus, Feb 24 2018
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A299824(n,f=exp(n),S=n/f,t)=for(j=2,oo,S+=(t=j^n*n^j)/(f*=j-1);tn&&return(ceil(S))) \\ For n > 23, use \p## with some ## >= 2n. - M. F. Hasler, Mar 09 2018
A346654
a(n) = Bell(2*n,n).
Original entry on oeis.org
1, 2, 94, 12351, 3188340, 1362057155, 869725707522, 775929767223352, 921839901090823112, 1406921223577401454239, 2682502220690005671884710, 6248503930824315386034050253, 17460431497766377837983159782652, 57647207262184459310081410522242310, 222006095854149044448961838142906736554
Offset: 0
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b:= proc(n, k) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
end:
a:= n-> b(2*n, n):
seq(a(n), n=0..14); # Alois P. Heinz, Jul 27 2021
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Table[BellB[2*n, n], {n, 0, 20}]
A346655
a(n) = Bell(3*n,n).
Original entry on oeis.org
1, 5, 2430, 5597643, 35618229364, 483040313859705, 11977437107679230274, 490630568583958198181583, 30889771581097736768046865352, 2832037863467651034046820871428061, 362579939205426756198837321528946171110, 62687814132880422794200073791149602981717667
Offset: 0
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b:= proc(n, k) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
end:
a:= n-> b(3*n, n):
seq(a(n), n=0..11); # Alois P. Heinz, Jul 27 2021
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Table[BellB[3*n, n], {n, 0, 15}]
A343263
a(0) = 1; a(n+1) = exp(-a(n)) * Sum_{k>=0} a(n)^k * k^n / k!.
Original entry on oeis.org
1, 1, 1, 2, 22, 301554, 2493675105669492542968967478
Offset: 0
-
a[0] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 1, k] a[n - 1]^k, {k, 0, n - 1}]; Table[a[n], {n, 0, 6}]
a[0] = 1; a[n_] := a[n] = BellB[n - 1, a[n - 1]]; Table[a[n], {n, 0, 6}]
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