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

%I A073400 #25 Nov 25 2024 20:44:31
%S A073400 2,9,33,45,396,831,243,3744,18297,28236,1377,32481,273483,968679,
%T A073400 1210140,8019,268029,3418767,20681811,58920534,62686440,47385,2130138,
%U A073400 38186478,347584284,1683064737,4075425738,3810867480
%N A073400 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073399.
%C A073400 The row polynomials are q(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,...
%C A073400 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 p(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073399(k,m).
%H A073400 Sean A. Irvine, <a href="/A073400/b073400.txt">Table of n, a(n) for n = 0..989</a>
%H A073400 Wolfdieter Lang, <a href="/A073399/a073399.txt">First 7 rows</a>.
%F A073400 Recursion for row polynomials defined in the comments: see A073401.
%e A073400 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 A073400 1; 9,33; 45,396,831; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).
%Y A073400 Cf. A001045, A073370, A073399, A073401.
%K A073400 nonn,easy,tabl
%O A073400 0,1
%A A073400 _Wolfdieter Lang_, Aug 02 2002