A054446 -id:A054446 - cp's OEIS frontend

A054447 Row sums of triangle A054446 (partial row sums triangle of Fibonacci convolution triangle).

1, 3, 9, 26, 73, 201, 545, 1460, 3873, 10191, 26633, 69198, 178889, 460437, 1180545, 3016552, 7684481, 19522203, 49473097, 125093506, 315654537, 795016545, 1998909985, 5017895196, 12578040097, 31485713511, 78716283081, 196563649718, 490301138569, 1221726409005

Offset: 0

Wolfdieter Lang, Apr 27 2000

Michael De Vlieger, Table of n, a(n) for n = 0..2604
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Index entries for linear recurrences with constant coefficients, signature (4, -2, -4, -1).

Cf. A000129, A000045, A054446, A001477.

LinearRecurrence[{4, -2, -4, -1}, {1, 3, 9, 26}, 30] (* Michael De Vlieger, Jun 23 2020 *)

a(n):=sum(k*sum(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k),k,1,n); /* Vladimir Kruchinin, Sep 06 2010 */

a(n) = Sum_{m=0..n} A054446(n,m) = ((n+1)*P(n+2)+(2-n)*P(n+1))/4, with P(n)=A000129(n) (Pell numbers).
G.f.: Pell(x)/(1-x*Fib(x)) = (Pell(x)^2)/Fib(x), with Pell(x)= 1/(1-2*x-x^2) = g.f. A000129(n+1) (Pell numbers without 0) and Fib(x)=1/(1-x-x^2) = g.f. A000045(n+1) (Fibonacci numbers without 0).
a(n) = Sum_(k*Sum_(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k),k,1,n), n>0. - Vladimir Kruchinin, Sep 06 2010
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4), a(0)=1, a(1)=3, a(2)=9, a(3)=26. - Philippe Deléham, Jan 22 2014
G.f.: (1-x-x^2)/(1-2*x-x^2)^2 = g(f(x))/x, where g is g.f. of A001477 and f is g.f. of A000045. - Oboifeng Dira, Jun 21 2020

A054448 Triangle of partial row sums of triangle A054446(n,m), n >= m >= 0.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 26, 14, 5, 1, 73, 44, 20, 6, 1, 201, 131, 69, 27, 7, 1, 545, 376, 220, 102, 35, 8, 1, 1460, 1052, 665, 349, 144, 44, 9, 1, 3873, 2888, 1937, 1116, 528, 196, 54, 10, 1, 10191, 7813, 5490, 3402, 1788, 768, 259, 65, 11, 1, 26633, 20892, 15240, 10008

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

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))^2)/(Fib(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).

This is the second member of the family of Riordan-type matrices obtained from the Fibonacci convolution matrix A037027 by repeated application of the partial row sums procedure.

			{1}; {3,1}; {9,4,1}; {26,14,5,1};...
Fourth row polynomial (n=3): p(3,x)= 26+14*x+5*x^2+x^3

Cf. A037027, A000045, A000129. Row sums: A054449.

a(n, m)=sum(A054446(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)*A037027(n-j, 0), j=m..n) + A054446(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (((Pell(x))^2)/Fib(x))*(x*Fib(x))^m, m >= 0, with Fib(x) = g.f. A000045(n+1) and Pell(x) = g.f. A000129(n+1).

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A054447 Row sums of triangle A054446 (partial row sums triangle of Fibonacci convolution triangle).

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