cp's OEIS Frontend

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

%I A073401 #30 Nov 25 2024 20:44:22
%S A073401 1,30,9,1050,531,63,44520,29610,6165,405,2245320,1789614,502821,59454,
%T A073401 2511,131891760,120133692,41182344,6686631,517104,15309,8862693840,
%U A073401 8966770308,3559509360,714174327,76790673,4214349,92583
%N A073401 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073402.
%C A073401 The row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...
%C A073401 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 q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073402(k,m).
%H A073401 Sean A. Irvine, <a href="/A073401/b073401.txt">Table of n, a(n) for n = 0..989</a>
%H A073401 Wolfdieter Lang, <a href="/A073401/a073401.txt">First 7 rows</a>.
%F A073401 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]
%e A073401 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.
%e A073401 1; 30,9; 1050,531,63; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).
%Y A073401 Cf. A001045, A073370, A073399, A073372, A073402, A073400.
%K A073401 nonn,easy,tabl
%O A073401 0,2
%A A073401 _Wolfdieter Lang_, Aug 02 2002