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A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!. 10
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 (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). 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

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

Sum_{k=0..n-1} A(n-k,k) = A005490(n). - Alois P. Heinz, Jan 05 2022

EXAMPLE

Array A(n,k) begins:

   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.

Main diagonal gives A134980.

Cf. A005490.

Antidiagonal sums give A347420.

Sequence in context: A110488 A271025 A134379 * A123158 A185414 A346520

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 August 11 15:10 EDT 2022. Contains 356066 sequences. (Running on oeis4.)