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A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)*binomial(-k,j-1)*x^j/j). 8
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 73, 120, 1, 1, 1, 6, 33, 154, 501, 720, 1, 1, 1, 7, 46, 273, 1306, 4051, 5040, 1, 1, 1, 8, 61, 436, 2721, 12976, 37633, 40320, 1, 1, 1, 9, 78, 649, 4956, 31701, 147484, 394353, 362880, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

FORMULA

Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} B(j,k)*A(n-j,k)/(n-j)! for n > 0.

EXAMPLE

Square array B(j,k) begins:

   1,   1,   1,    1,    1, ...

   0,   1,   2,    3,    4, ...

   0,   1,   3,    6,   10, ...

   0,   1,   4,   10,   20, ...

   0,   1,   5,   15,   35, ...

   0,   1,   6,   21,   56, ...

Square array A(n,k) begins:

   1,   1,   1,    1,    1, ...

   1,   1,   1,    1,    1, ...

   1,   2,   3,    4,    5, ...

   1,   6,  13,   22,   33, ...

   1,  24,  73,  154,  273, ...

   1, 120, 501, 1306, 2721, ...

MATHEMATICA

B[j_, k_] := (-1)^(j-1)*Binomial[-k, j-1];

A[0, _] = 1; A[n_, k_] := (n-1)!*Sum[B[j, k]*A[n-j, k]/(n-j)!, {j, 1, n}];

Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Nov 03 2017 *)

CROSSREFS

Columns k=0..10 give A000012, A000142, A000262, A049376, A049377, A049378, A049402, A132164, A293986, A293987, A293988.

Rows n=0-1 give A000012.

Main diagonal gives A293989.

Cf. A293012, A293991.

Sequence in context: A275043 A227061 A201949 * A326323 A257493 A296526

Adjacent sequences:  A291706 A291707 A291708 * A291710 A291711 A291712

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Oct 21 2017

STATUS

approved

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Last modified July 14 16:21 EDT 2020. Contains 335729 sequences. (Running on oeis4.)