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A073405 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n >= 0, (generalized (2,2)-Fibonacci). Companion triangle is A073406.

%I A073405 #20 Nov 25 2024 21:19:23
%S A073405 1,36,12,1536,888,120,80448,62592,15168,1152,5068800,4813056,1600704,
%T A073405 222336,10944,375598080,413351424,169917696,32811264,2992896,103680,
%U A073405 32103751680,39661608960,19066503168,4592982528
%N A073405 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n >= 0, (generalized (2,2)-Fibonacci). Companion triangle is A073406.
%C A073405 The row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...
%C A073405 The k-th convolution of U0(n) := A002605(n), n >= 0, ((2,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073387(n+k,k) = 2*(p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*U0(n))/(k!*12^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)= A073406(k,m).
%H A073405 W. Lang, <a href="/A073405/a073405_6.txt">First 7 rows</a>.
%F A073405 Recursion for row polynomials defined in the comments: p(k, n)= 2*(2*(n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n+1)); q(k, n)= 4*((n+1)*p(k-1, n+1)+(n+2*(k+1))*q(k-1, n)), k >= 1. [Corrected by _Sean A. Irvine_, Nov 25 2024]
%e A073405 k=2: U2(n)=2*((36+12*n)*(n+1)*U0(n+1)+(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
%e A073405 Triangle begins:
%e A073405   1;
%e A073405   36, 12;
%e A073405   1536, 888, 120;
%e A073405   ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).
%Y A073405 Cf. A002605, A073387, A073406, A073403.
%K A073405 nonn,easy,tabl
%O A073405 0,2
%A A073405 _Wolfdieter Lang_, Aug 02 2002