A058402 Coefficient triangle of polynomials (rising powers) related to Pell number convolutions. Companion triangle is A058403.

1, 22, 8, 588, 376, 56, 19656, 17024, 4576, 384, 801360, 848096, 313504, 48256, 2624, 38797920, 47494272, 21685888, 4643072, 468608, 17920, 2181332160, 2986217856, 1590913920, 424509952, 60136448, 4307456, 122368, 139864717440

Offset

0,2

Comments

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

The k-th convolution of P0(n) := A000129(n+1) n >= 0, (Pell numbers starting with P0(0)=1) with itself is Pk( n) := A054456(n+k,k) = (p(k-1,n)*(n+1)*2*P0(n+1) + q(k-1,n)*(n+2)*P0(n))/(k!*8^k)), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A058403(k,m).

Links

Table of n, a(n) for n=0..28.

W. Lang, First 7 rows, also for A058403.

Formula

Recursion for row polynomials defined in the comments: p(k, n)= 4*(n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n); q(k, n)= 4*(n+1)*p(k-1, n+1)+2*(n+2*(k+1))*q(k-1, n+1), k >= 1. [Corrected by Sean A. Irvine, Aug 05 2022]

Example

k=2: P2(n)=((22+8*n)*(n+1)*2*P0(n+1)+(20+8*n)*(n+2)*P0(n))/128, cf. A054457.
1; 22,8; 588,376,56; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Crossrefs

Cf. A000129, A054456, A058403, A058404-5 (falling powers).

Sequence in context: A298176 A113783 A298781 * A248032 A068611 A104061

Adjacent sequences: A058399 A058400 A058401 * A058403 A058404 A058405

Keyword

nonn,tabl

Author

Wolfdieter Lang, Dec 11 2000

Extensions

Link and cross-references added by Wolfdieter Lang, Jul 31 2002

Status

approved