A189233 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals upwards, where the e.g.f. of column k is exp(k*(e^x-1)).
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 94, 57, 20, 5, 1, 0, 203, 454, 309, 116, 30, 6, 1, 0, 877, 2430, 1866, 756, 205, 42, 7, 1, 0, 4140, 14214, 12351, 5428, 1555, 330, 56, 8, 1, 0, 21147, 89918, 88563, 42356, 12880, 2850, 497, 72, 9, 1
Offset: 0
Examples
Square array begins:
A000007 A000110 A001861 A027710 A078944 A144180 A144223 A144263
A000012 1, 1, 1, 1, 1, 1, 1, 1, ...
A001477 0, 1, 2, 3, 4, 5, 6, 7, ...
A002378 0, 2, 6, 12, 20, 30, 42, 56, ...
A033445 0, 5, 22, 57, 116, 205, 330, 497, ...
0, 15, 94, 309, 756, 1555, 2850, 4809, ...
0, 52, 454, 1866, 5428, 12880, 26682, 50134, ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..5150
- E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
- Peter Luschny, Set partitions and Bell numbers
Crossrefs
Programs
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Maple
# Cf. also the Maple prog. of Alois P. Heinz in A144223 and A144180. expnums := proc(k,n) option remember; local j; `if`(n = 0, 1, (1+add(binomial(n-1,j-1)*expnums(k,n-j), j = 1..n-1))*k) end: A189233_array := (k, n) -> expnums(k,n): seq(print(seq(A189233_array(k,n), k = 0..7)), n = 0..5); A189233_egf := k -> exp(k*(exp(x)-1)); T := (n,k) -> n!*coeff(series(A189233_egf(k), x, n+1), x, n): seq(lprint(seq(T(n,k), k = 0..7)), n = 0..5): # alternative Maple program: 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(d-k, k), k=0..d), d=0..12); # Alois P. Heinz, Sep 25 2017
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Mathematica
max = 9; Clear[col]; col[k_] := col[k] = CoefficientList[ Series[ Exp[k*(Exp[x]-1)], {x, 0, max}], x]*Range[0, max]!; a[0, ] = 1; a[n?Positive, 0] = 0; a[n_, k_] := col[k][[n+1]]; Table[ a[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *) Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid (* Geoffrey Critzer, Dec 23 2018 *) -
Maxima
A(n,k):=if k=0 and n=0 then 1 else if k=0 then 0 else sum(stirling2(n,i)*k^i,i,0,n); /* Vladimir Kruchinin, Apr 12 2019 */
Formula
E.g.f. of column k: exp(k*(e^x-1)).
A(n,k) = BellPolynomial(n, k). - Geoffrey Critzer, Dec 23 2018
A(n,k) = Sum_{i=0..n} Stirling2(n,i)*k^i. - Vladimir Kruchinin, Apr 12 2019
Comments