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A058402 Coefficient triangle of polynomials (rising powers) related to Pell number convolutions. Companion triangle is A058403. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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), k >= 1.

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

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Last modified October 16 05:25 EDT 2021. Contains 348035 sequences. (Running on oeis4.)