A029656 Numbers in the (2,1)-Pascal triangle A029653 that are different from 1.

2, 2, 3, 2, 5, 4, 2, 7, 9, 5, 2, 9, 16, 14, 6, 2, 11, 25, 30, 20, 7, 2, 13, 36, 55, 50, 27, 8, 2, 15, 49, 91, 105, 77, 35, 9, 2, 17, 64, 140, 196, 182, 112, 44, 10, 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 2, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 2, 23, 121, 385

Offset

1,1

References

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; triangle on page 6, numerators.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150)

Eric Weisstein's World of Mathematics, Alternating Sign Matrix.

D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954.

Formula

From Thomas Baruchel, Jun 26 2018: (Start)

a(n,k) = (binomial(n+2,k+1) + binomial(n+1,k) + binomial(n,k) - binomial(n,k+1))/2.

a(n,k) = binomial(n-1,k-1) + binomial(n-1,k) + binomial(n,k-1) + binomial(n,k). (End)

Example

Triangle begins:
  2;
  2,  3;
  2,  5,  4;
  2,  7,  9,  5;
  2,  9, 16, 14,  6;
  2, 11, 25, 30, 20,  7;
  ...

Mathematica

Table[(Binomial[n + 2, k + 1] + Binomial[n + 1, k] + Binomial[n, k] - Binomial[n, k + 1])/2, {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 29 2018 *)

Crossrefs

Cf. A048601, A029638.

Sequence in context: A307687 A281121 A220949 * A121306 A073311 A347560

Adjacent sequences: A029653 A029654 A029655 * A029657 A029658 A029659

Keyword

nonn,tabl

Author

Mohammad K. Azarian

Extensions

More terms from James A. Sellers

Status

approved