A058402 - OEIS

%I #12 Aug 05 2022 22:27:18

%S 1,22,8,588,376,56,19656,17024,4576,384,801360,848096,313504,48256,

%T 2624,38797920,47494272,21685888,4643072,468608,17920,2181332160,

%U 2986217856,1590913920,424509952,60136448,4307456,122368,139864717440

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

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

%C 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).

%H W. Lang, <a href="/A058402/a058402_3.txt">First 7 rows, also for A058403</a>.

%F 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]

%e k=2: P2(n)=((22+8*n)*(n+1)*2*P0(n+1)+(20+8*n)*(n+2)*P0(n))/128, cf. A054457.

%e 1; 22,8; 588,376,56; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

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

%K nonn,tabl

%O 0,2

%A _Wolfdieter Lang_, Dec 11 2000

%E Link and cross-references added by _Wolfdieter Lang_, Jul 31 2002