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A058403 Coefficient triangle of polynomials (rising powers) related to Pell number convolutions. Companion triangle is A058402. 3

%I #8 Aug 29 2019 13:13:19

%S 2,20,8,360,288,48,9840,11360,3520,320,363360,522752,225344,37888,

%T 2176,16776000,27849600,14871296,3491072,373504,14848,922158720,

%U 1692808704,1053556480,308703232,46459904,3467264,101376,58499239680,115821927936

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

%C The row polynomials are q(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 p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A058402(k,m).

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

%F Recursion for row polynomials defined in the comments: see A058402.

%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 2; 20,8; 360,288,48; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

%Y Cf. A000129, A054456, A058402, A058404-5 (falling powers).

%K nonn,tabl

%O 0,1

%A _Wolfdieter Lang_, Dec 11 2000

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

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Last modified March 28 21:54 EDT 2024. Contains 371254 sequences. (Running on oeis4.)