A114213 A generalized Pascal triangle modulo 2.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1

Offset

0,1

Comments

Row sums are A114212. Diagonal sums are A114214.

Row sums of inverse are 0^n (conjecture).

Links

Table of n, a(n) for n=0..104.

Jeffrey Shallit and Lukas Spiegelhofer, Continuants, run lengths, and Barry's modified Pascal triangle, arXiv:1710.06203 [math.CO], 2017.

Formula

T(n, k) = (Sum_{j=0..n-k} C(k, j)*C(n-k, j)*(1+(-1)^j)/2) mod 2.

Example

Triangle begins
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1, 1;
  1, 1, 0, 1, 1;
  1, 1, 0, 0, 1, 1;
  1, 1, 1, 0, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1;
  1, 1, 0, 1, 0, 1, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 0, 0, 1, 1;
  1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1;

Prog

(PARI) T(n, k) = sum(j=0, n-k, binomial(k, j)*binomial(n-k, j)*(1+(-1)^j)/2) % 2; \\ Michel Marcus, Jun 06 2021

Crossrefs

Cf. A114212, A114214.

Sequence in context: A178788 A131217 A105567 * A108358 A267959 A144384

Adjacent sequences: A114210 A114211 A114212 * A114214 A114215 A114216

Keyword

easy,nonn,tabl

Author

Paul Barry, Nov 17 2005

Status

approved