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

0,2

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.

Links

Table of n, a(n) for n=0..65.

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.

Formula

a(n, m)=sum(A054450(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in column m).

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<m.

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).

Example

{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

Crossrefs

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

Sequence in context: A106471 A180870 A228565 * A109433 A123490 A157028

Adjacent sequences: A054450 A054451 A054452 * A054454 A054455 A054456

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang, Apr 27 2000 and May 08 2000.

Status

approved