A115150 -id:A115150 - cp's OEIS frontend

A113647 Triangle of numbers related to the generalized Catalan sequence C(2;n+1)=A064062(n+1), n>=0.

1, 1, 3, 1, 7, 13, 1, 15, 41, 67, 1, 31, 113, 247, 381, 1, 63, 289, 783, 1545, 2307, 1, 127, 705, 2271, 5361, 9975, 14589, 1, 255, 1665, 6207, 16929, 36879, 66057, 95235, 1, 511, 3841, 16255, 50113, 123871, 255985, 446455, 636925, 1, 1023, 8705, 41215, 141441

Offset: 0

Wolfdieter Lang, Jan 13 2006

This triangle, called Y(2,1), appears in the totally asymmetric exclusion process for the (unphysical) values alpha=2, beta=1. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
The main diagonal (M=1) gives the generalized Catalan sequence C(2,n):=A064062(n).
The diagonal sequences give A064062(n+1), A115137, A115150-A115153, for n+1>= M=1,..,6.

			Triangle begins:
  1;
  1,3;
  1,7,13;
  1,15,41,67;
  1,31,113,247,381;
  ...
113=a(4,3)= a(4,2) + 2*a(3,3)= 31 + 2*41.

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
Wolfdieter Lang, First 10 rows.

Row sums give A115136.

a(n, n+1)=A064062(n+1) (main diagonal with M=1); a(n, n-M+2)= a(n, n-M+1) + 2*a(n-1, n-M+2), M>=2; a(n, 1)=1; n>=0.
G.f. for diagonal sequence M=1: GY(1, x):=(2*c(2*x)-1)/(1+x) with c(x) g.f. of A000108 (Catalan); for M=2: GY(2, x)=(1-2*x)*GY(1, x)-1; for M>=3: GY(M, x)= GY(M-1, x) -2*x*GY(M-2, x) + x^(M-2).
G.f. for diagonal sequence M (solution to the above given recurrence): GY(M, x)= (x^(M-1)/(1+x))*( 2^(M+1)*x*(p(M, 2*x)-(2*x)*p(M+1, 2*x)*c(2*x))+1), with c(x) g.f. of A000108 (Catalan) and p(n, x):= -((1/sqrt(x))^(n+1))*S(n-1, 1/sqrt(x)) with Chebyshev's S(n, x) polynomials given in A049310.

A115138 A sequence related to Catalan numbers A000108.

Original entry on oeis.org

1, -1, 1, 15, 113, 783, 5361, 36879, 255985, 1794063, 12689393, 90505231, 650379249, 4705157135, 34244198385, 250572963855, 1842382110705, 13605619630095, 100872203796465, 750556607938575, 5602962592235505, 41952165966643215, 314983352736153585

Offset: 0

Wolfdieter Lang, Jan 13 2006

See also A115150, the third diagonal of triangle A113647 (called Y(2,1)).

			15= a(3) = A062992(3) - 4*A062992(2) = 67 - 4*13.

(1-4*x)*(2*c(2*x)-1)/(1+x) with c(x) g.f. of A000108 (Catalan).
a(n)= A113647(n, n-1), n>=2.
a(n)= b(n)-4*b(n-1) with b(n):=A062992(n), n>=2; a(0)=1, a(1)=-1.

cp's OEIS Frontend

A113647 Triangle of numbers related to the generalized Catalan sequence C(2;n+1)=A064062(n+1), n>=0.

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