A381567 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381568.

1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 14, 22, 0, 1, 8, 27, 64, 126, 0, 1, 10, 44, 134, 365, 884, 0, 1, 12, 65, 240, 777, 2492, 7149, 0, 1, 14, 90, 390, 1438, 5238, 19578, 64688, 0, 1, 16, 119, 592, 2440, 9696, 40244, 172356, 641836, 0, 1, 18, 152, 854, 3891, 16632, 73408, 345726, 1668686, 6888740, 0

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+2*k,j)/(n-j+k) * A(n-j,j).

Example

Square array begins:
  1,    1,     1,     1,     1,      1,      1, ...
  0,    2,     4,     6,     8,     10,     12, ...
  0,    5,    14,    27,    44,     65,     90, ...
  0,   22,    64,   134,   240,    390,    592, ...
  0,  126,   365,   777,  1438,   2440,   3891, ...
  0,  884,  2492,  5238,  9696,  16632,  27036, ...
  0, 7149, 19578, 40244, 73408, 125035, 203258, ...

Prog

(PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+2*k, j)/(n-j+k)*a(n-j, j)));

Crossrefs

Columns k=0..1 give A000007, A381568.

Cf. A381566, A381569.

Sequence in context: A139435 A077909 A247126 * A342134 A349740 A327117

Adjacent sequences: A381564 A381565 A381566 * A381568 A381569 A381570

Keyword

nonn,tabl,new

Author

Seiichi Manyama, Feb 28 2025

Status

approved