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A108087
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Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.
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11
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1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 10, 4, 1, 52, 52, 37, 17, 5, 1, 203, 203, 151, 77, 26, 6, 1, 877, 877, 674, 372, 141, 37, 7, 1, 4140, 4140, 3263, 1915, 799, 235, 50, 8, 1, 21147, 21147, 17007, 10481, 4736, 1540, 365, 65, 9, 1, 115975, 115975, 94828, 60814, 29371, 10427, 2727, 537, 82, 10, 1
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OFFSET
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0,4
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COMMENTS
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The column for k=0 is A000110 (Bell or exponential numbers). The column for k=1 is A000110 starting at offset 1. The column for k=2 is A005493 (Sum_{k=0..n} k*Stirling2(n,k).). The column for k=3 is A005494 (E.g.f.: exp(3*z+exp(z)-1). The column for k=4 is A045379 (E.g.f.: exp(4*z+exp(z)-1).). The row for n=0 is 1's sequence, the row for n=1 is the natural numbers. The row for n=2 is A002522 (n^2 + 1.). The row for n=3 is A005491 (n^3 + 3n + 1.). The row for n=4 is A005492.
Number of ways of placing n labeled balls into n+k boxes, where k of the boxes are labeled and the rest are indistinguishable. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
The column for k = -1 (not shown) is A000296 (Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.). - Gerald McGarvey, Oct 08 2006
Equals antidiagonals of an array in which (n+1)-th column is the binomial transform of n-th column, with leftmost column = the Bell sequence, A000110. - Gary W. Adamson, Apr 16 2009
Number of partitions of [n+k] where at least k blocks contain their own index element. A(2,2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, Jan 07 2022
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REFERENCES
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F. Ruskey, Combinatorial Generation, preprint, 2001.
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LINKS
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FORMULA
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For n> 1, A(n, k) = k^n + sum_{i=0..n-2} A086659(n, i)*k^i. (A086659 is set partitions of n containing k-1 blocks of length 1, with e.g.f: exp(x*y)*(exp(exp(x)-1-x)-1).)
A(n, k) = k * A(n-1, k) + A(n-1, k+1), A(0, k) = 1. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
A(n,k) = Sum_{i=0..n} C(n,i) * k^i * Bell(n-i). - Alois P. Heinz, Jul 18 2012
Sum_{k=0..n} T(n, k) = A347420(n). (End)
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EXAMPLE
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Array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... A000027;
2, 5, 10, 17, 26, 37, 50, 65, 82, 101, ... A002522;
5, 15, 37, 77, 141, 235, 365, 537, 757, 1031, ... A005491;
15, 52, 151, 372, 799, 1540, 2727, 4516, 7087, 10644, ... A005492;
52, 203, 674, 1915, 4736, 10427, 20878, 38699, 67340, 111211, ... ;
Antidiagonal triangle, T(n, k), begins as:
1;
1, 1;
2, 2, 1;
5, 5, 3, 1;
15, 15, 10, 4, 1;
52, 52, 37, 17, 5, 1;
203, 203, 151, 77, 26, 6, 1;
877, 877, 674, 372, 141, 37, 7, 1;
4140, 4140, 3263, 1915, 799, 235, 50, 8, 1;
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MAPLE
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with(combinat):
A:= (n, k)-> add(binomial(n, i) * k^i * bell(n-i), i=0..n):
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MATHEMATICA
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Unprotect[Power]; 0^0 = 1; A[n_, k_] := Sum[Binomial[n, i] * k^i * BellB[n - i], {i, 0, n}]; Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)
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PROG
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(PARI) f(n, k)=round (suminf(i=0, (i+k)^n/i!)/exp(1));
g(n, k)=for(k=0, k, print1(f(n, k), ", ")) \\ prints k+1 terms of n-th row
(Magma)
A108087:= func< n, k | (&+[Binomial(n-k, j)*k^j*Bell(n-k-j): j in [0..n-k]]) >;
(SageMath)
def A108087(n, k): return sum( k^j*bell_number(n-k-j)*binomial(n-k, j) for j in range(n-k+1))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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