A073402 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073401.

2, 33, 9, 831, 396, 45, 28236, 18297, 3744, 243, 1210140, 968679, 273483, 32481, 1377, 62686440, 58920534, 20681811, 3418767, 268029, 8019, 3810867480, 4075425738, 1683064737, 347584284, 38186478, 2130138, 47385

Offset: 0

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy
• tabl

Comments

The row polynomials are q(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of U0(n) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^k)), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073401(k,m).

Examples

			k=2: U2(n)=((30+9*n)*(n+1)*U0(n+1)+(33+9*n)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 33,9; 831,396,45; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).
		

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..989
• Wolfdieter Lang, First 7 rows.

Crossrefs

Cf. A001045, A073370, A073399, A073372, A073400.

Formula

Recursion for row polynomials defined in the comments: p(k, n)= (n+2)*p(k-1, n+1)+4*(n+2*(k+1))*p(k-1, n)+2*(n+3)*q(k-1, n+1); q(k, n)= (n+1)*p(k-1, n+1)+4*(n+2*(k+1))*q(k-1, n), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]