A395545 Square array A(n,k), n>=1, k>=1, read by descending antidiagonals A(n,k) = Sum_{i=0..k} binomial(n-i, i)*binomial(k+i, 2*i+1).

1, 1, 2, 1, 3, 4, 1, 4, 7, 6, 1, 5, 11, 13, 9, 1, 6, 16, 24, 22, 12, 1, 7, 22, 40, 46, 34, 16, 1, 8, 29, 62, 86, 80, 50, 20, 1, 9, 37, 91, 148, 166, 130, 70, 25, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 1, 11, 56, 174, 367, 553, 610, 496, 295, 125, 36

Offset

1,3

Links

Table of n, a(n) for n=1..66.

Flavien Mabilat, Une curieuse égalité entre deux sommes de produits de coefficients binomiaux, arXiv:2604.23223 [math.CO], 2026. See 2.4 p. 10.

Formula

A(n,k) = Sum_{i=0..k} binomial(n-i, i-1)*binomial(k+i, 2*i).

Example

Array begins:
  1,  1,  1,  1, ...
  2,  3,  4,  5, ...
  4,  7, 11, 16, ...
  6, 13, 24, 40, ...
  9, 22, 46, 86, ...
  ...

Prog

(PARI) A(n, k) = sum(i=0, k, binomial(n-i, i)*binomial(k+i, 2*i+1));

Crossrefs

Cf. A000012, A000027, A000124, A003600, A223718, ...

Cf. A002620, A002623, A001752, A001753, A001769, ...

Sequence in context: A132110 A039912 A163311 * A210555 A008949 A076832

Adjacent sequences: A395542 A395543 A395544 * A395546 A395547 A395548

Keyword

nonn,tabl

Author

Michel Marcus, Apr 28 2026

Status

approved