A054446 Triangle of partial row sums of triangle A037027(n,m), n >= m >= 0 (Fibonacci convolution triangle).

1, 2, 1, 5, 3, 1, 12, 9, 4, 1, 29, 24, 14, 5, 1, 70, 62, 42, 20, 6, 1, 169, 156, 118, 67, 27, 7, 1, 408, 387, 316, 205, 100, 35, 8, 1, 985, 951, 821, 588, 332, 142, 44, 9, 1, 2378, 2323, 2088, 1614, 1020, 509, 194, 54, 10, 1, 5741, 5652, 5232, 4290, 2966, 1671, 747, 257

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 Pell(z)/(1-x*z*Fib(z)) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0) and Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

Links

Table of n, a(n) for n=0..62.

Formula

a(n, m)=sum(A037027(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in columns m).

Column m recursion: a(n, m)= sum(a(j-1, m)*A037027(n-j, 0), j=m..n) + A037027(n, m), n >= m >= 0, a(n, m) := 0 if n<m.

G.f. for column m: Pell(x)*(x*Fib(x))^m, m >= 0, with Fib(x) = g.f. A000045(n+1) and Pell(x) = g.f. A000129(n+1).

T(n,0) = 2*T(n-1,0) + T(n-2,0), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k) for k>0, T(0,0) = 1, T(1,0) = 2, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 26 2014

Example

{1}; {2,1}; {5,3,1}; {12,9,4,1};...
Fourth row polynomial (n=3): p(3,x)= 12+9*x+4*x^2+x^3

Crossrefs

Cf. A037027, A000045, A000129. Row sums: A054447(n).

Sequence in context: A327631 A130197 A106513 * A164981 A366858 A047858

Adjacent sequences: A054443 A054444 A054445 * A054447 A054448 A054449

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang, Apr 27 2000 and May 08 2000

Status

approved