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A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0}(i+k)^n/i!. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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). From expansion of falling factorials (binomial transform of A005493)). 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 (From expansion of falling factorials.).

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

REFERENCES

F. Ruskey, Combinatorial Generation, preprint, 2001.

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1, Figure 1.

J. Riordan, Letter, Oct 31 1977, The array is on the second page.

F. Ruskey, Combinatorial Generation, 2003

F. Ruskey, Lexicographic Algorithms

FORMULA

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

EXAMPLE

   1,     1,     1,     1,     1,     1,     1,     1,     1,     1,     1, ...

     1,     2,     3,     4,     5,     6,     7,     8,     9,    10,    11,...

     2,     5,    10,    17,    26,    37,    50,    65,    82,   101,   122,...

     5,    15,    37,    77,   141,   235,   365,   537,   757,  1031,  1365,...

    15,    52,   151,   372,   799,  1540,  2727,  4516,  7087, 10644, 15415,...

    52,   203,   674,  1915,  4736, 10427, 20878, 38699, 67340,111211,175802,...

MAPLE

with(combinat):

A:= (n, k)-> add(binomial(n, i) * k^i * bell(n-i), i=0..n):

seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 18 2012

MATHEMATICA

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-Fran├žois Alcover, Nov 05 2015, after Alois P. Heinz *)

PROG

(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

CROSSREFS

Cf. A000110, A005493, A005494, A045379, A002522, A005491, A005492, A086659, A134980 (diagonal)

Sequence in context: A110488 A271025 A134379 * A123158 A185414 A133611

Adjacent sequences:  A108084 A108085 A108086 * A108088 A108089 A108090

KEYWORD

nonn,tabl

AUTHOR

Gerald McGarvey, Jun 05 2005

STATUS

approved

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Last modified May 23 11:05 EDT 2019. Contains 323513 sequences. (Running on oeis4.)