The On-Line Encyclopedia of Integer Sequences (OEIS)

A306245 revision #77

A306245

Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} k^j * binomial(n-1,j) * A(j,k) for n > 0.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 17, 15, 1, 1, 1, 5, 43, 179, 52, 1, 1, 1, 6, 89, 1279, 3489, 203, 1, 1, 1, 7, 161, 5949, 108472, 127459, 877, 1, 1, 1, 8, 265, 20591, 1546225, 26888677, 8873137, 4140, 1

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OFFSET

0,9

LINKS

Seiichi Manyama, Antidiagonals n = 0..55, flattened

FORMULA

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(k * x / (1 - x)) / (1 - x). - Seiichi Manyama, Jun 18 2022

EXAMPLE

Square array begins:

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

1, 2, 3, 4, 5, 6, ...

1, 5, 17, 43, 89, 161, ...

1, 15, 179, 1279, 5949, 20591, ...

1, 52, 3489, 108472, 1546225, 12950796, ...

MAPLE

A:= proc(n, k) option remember; `if`(n=0, 1,

add(k^j*binomial(n-1, j)*A(j, k), j=0..n-1))

end:

seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Jul 28 2019

MATHEMATICA

A[0, _] = 1;

A[n_, k_] := A[n, k] = Sum[k^j Binomial[n-1, j] A[j, k], {j, 0, n-1}];

Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 29 2020 *)

CROSSREFS

Columns k=0..4 give A000012, A000110, A126443, A355081, A355082.

Rows n=0+1, 2 give A000012, A000027(n+1).

Main diagonal gives A309401.

Cf. A309386.

Sequence in context: A124560 A368025 A290759 * A275043 A227061 A201949

Adjacent sequences: A306242 A306243 A306244 * A306246 A306247 A306248

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Jul 28 2019

STATUS

editing