A054453 Triangle of partial row sums of triangle A054450(n,m), n >= m >= 0.

1, 2, 1, 4, 2, 1, 8, 5, 2, 1, 15, 10, 6, 2, 1, 28, 20, 12, 7, 2, 1, 51, 38, 26, 14, 8, 2, 1, 92, 71, 50, 33, 16, 9, 2, 1, 164, 130, 97, 64, 41, 18, 10, 2, 1, 290, 235, 180, 130, 80, 50, 20, 11, 2, 1, 509, 420, 332, 244, 171, 98, 60, 22, 12, 2, 1

Offset: 0

Author

Wolfdieter Lang, Apr 27 2000 and May 08 2000

Keywords

• easy
• nonn
• tabl

Comments

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z^2)*(Fib(z))^2)/(1-x*z/(1-z^2)) Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

This is the second member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310(n,m) by repeated application of the partial row sums procedure.

The column sequences are A029907, A001629, A054454 for m=0..2.

Examples

			{1}; {2,1}; {4,2,1}; {8,5,2,1};...
Fourth row polynomial (n=3): p(3,x)= 8+5*x+2*x^2+x^3
		

Links

• Gregg Musiker, Nick Ovenhouse, and Sylvester W. Zhang, Double Dimers and Super Ptolemy Relations, Séminaire Lotharingien de Combinatoire XX, Proc. 35th Conf. Formal Power, Series and Algebraic Combinatorics (Davis) 2023, Art. #YY. See p. 12.

Crossrefs

Cf. A049310, A054450, A000045, A029907, A001629. Row sums: A054455(n).

Formula

a(n, m)=sum(A054450(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j-1, m)*|A049310(n-j, 0)|, j=m..n) + A054450(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: ((1-x^2)*(Fib(x))^2)*(x/(1-x^2))^m, m >= 0, with Fib(x) G.f. for A000045(n+1).

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