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A073399 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n>=0, (generalized (1,2)-Fibonacci). Companion triangle is A073400.

%I A073399 #30 Nov 25 2024 20:44:35
%S A073399 1,9,30,63,531,1050,405,6165,29610,44520,2511,59454,502821,1789614,
%T A073399 2245320,15309,517104,6686631,41182344,120133692,131891760,92583,
%U A073399 4214349,76790673,714174327,3559509360,8966770308,8862693840
%N A073399 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n>=0, (generalized (1,2)-Fibonacci). Companion triangle is A073400.
%C A073399 The row polynomials are p(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..
%C A073399 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^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073400(k,m).
%H A073399 Sean A. Irvine, <a href="/A073399/b073399.txt">Table of n, a(n) for n = 0..989</a>
%H A073399 Wolfdieter Lang, <a href="/A073399/a073399.txt">First 7 rows.</a>
%F A073399 Recursion for row polynomials defined in the comments: see A073401.
%e A073399 k=2: U2(n)=((9*n+30)*(n+1)*U0(n+1)+(9*n+33)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
%e A073399 1; 9,30; 63,531,1050; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).
%Y A073399 Cf. A001045, A073370, A073400-A073402, A073372.
%K A073399 nonn,easy,tabl
%O A073399 0,2
%A A073399 _Wolfdieter Lang_, Aug 02 2002