A292860
Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 15, 0, 1, 5, 20, 57, 94, 52, 0, 1, 6, 30, 116, 309, 454, 203, 0, 1, 7, 42, 205, 756, 1866, 2430, 877, 0, 1, 8, 56, 330, 1555, 5428, 12351, 14214, 4140, 0, 1, 9, 72, 497, 2850, 12880, 42356, 88563, 89918, 21147, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 6, 12, 20, 30, 42, ...
0, 5, 22, 57, 116, 205, 330, ...
0, 15, 94, 309, 756, 1555, 2850, ...
0, 52, 454, 1866, 5428, 12880, 26682, ...
0, 203, 2430, 12351, 42356, 115155, 268098, ...
Columns k=0-10 give:
A000007,
A000110,
A001861,
A027710,
A078944,
A144180,
A144223,
A144263,
A221159,
A276506,
A276507.
Same array, different indexing is
A189233.
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A:= proc(n, k) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
end:
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 25 2017
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A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[, ] = 0;
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)
A221176
a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(4i).
Original entry on oeis.org
1, 16, 272, 4880, 91920, 1810192, 37142288, 791744272, 17490370320, 399558315792, 9421351690000, 228916588400400, 5723078052339472, 147025755978698512, 3876566243300318992, 104789417805394595600, 2901159958960121863952, 82188946843192555474704, 2380551266738846355103504, 70441182699006212824911632
Offset: 0
-
With[{nn=20},CoefficientList[Series[Exp[16 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 19 2024 *)
A344840
a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(k-1).
Original entry on oeis.org
1, 5, 35, 265, 2195, 19625, 187755, 1909185, 20521515, 232124745, 2752591475, 34108980105, 440444019835, 5912197332865, 82320781521195, 1186703083508025, 17680850448587155, 271845880552898985, 4307188044378111915, 70236616096770062945, 1177406236243423738475
Offset: 0
-
a[0] = 1; a[n_] := a[n] = 5 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1 + 5 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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
A276506
E.g.f.: exp(9*(exp(x)-1)).
Original entry on oeis.org
1, 9, 90, 981, 11511, 144108, 1911771, 26730981, 392209380, 6016681467, 96202473183, 1599000785730, 27563715220509, 491777630207037, 9064781481234546, 172346601006842337, 3375007346801025099, 67983454804021156548, 1406921223577401454239, 29881379179971835132761
Offset: 0
-
a:= proc(n) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*9)
end:
seq(a(n), n=0..25); # Alois P. Heinz, Sep 25 2017
-
Table[BellB[n, 9], {n, 0, 30}]
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my(x='x+O('x^99)); Vec(serlaplace(exp(9*(exp(x)-1)))) \\ Altug Alkan, Sep 17 2016
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