A054453 - OEIS

%I #11 Apr 18 2023 11:57:04

%S 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,

%T 71,50,33,16,9,2,1,164,130,97,64,41,18,10,2,1,290,235,180,130,80,50,

%U 20,11,2,1,509,420,332,244,171,98,60,22,12,2,1

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

%C 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).

%C 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.

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

%H Gregg Musiker, Nick Ovenhouse, and Sylvester W. Zhang, <a href="https://www-users.cse.umn.edu/~swzhang/files/MOZ-FPSAC23.pdf">Double Dimers and Super Ptolemy Relations</a>, Séminaire Lotharingien de Combinatoire XX, Proc. 35th Conf. Formal Power, Series and Algebraic Combinatorics (Davis) 2023, Art. #YY. See p. 12.

%F 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).

%F 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.

%F 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).

%e {1}; {2,1}; {4,2,1}; {8,5,2,1};...

%e Fourth row polynomial (n=3): p(3,x)= 8+5*x+2*x^2+x^3

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

%K easy,nonn,tabl

%O 0,2

%A _Wolfdieter Lang_, Apr 27 2000 and May 08 2000.