cp's OEIS Frontend

A187537 Riordan array (1, (A000045(x)/x-1) *A001006(A000045(x)/x-1) ).

%I A187537 #19 Feb 13 2023 11:29:27
%S A187537 1,3,1,9,6,1,31,27,9,1,113,116,54,12,1,431,493,282,90,15,1,1697,2098,
%T A187537 1383,556,135,18,1,6847,8975,6567,3107,965,189,21,1,28161,38640,30636,
%U A187537 16376,6070,1536,252,24,1
%N A187537 Riordan array (1, (A000045(x)/x-1) *A001006(A000045(x)/x-1) ).
%C A187537 The column with index 0 of the standard array is not incorporated in this triangle. (It contains a 1 followed by zeros.)
%C A187537 The truncated Fibonacci sequence is A000045(x)/x-1 = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5+ ...
%C A187537 The composition with the Motzkin sequence is A001006(...) = 1 + x + 4*x^2 + 15*x^3 + 58*x^4 + 229*x^5 + ...
%C A187537 Eventually this defines the second component in the definition (A000045(...)/x-1)*A001006(...) = x + 3*x^2 + 9*x^3 + 31*x^4 + 113*x^5 + 431*x^6 + ... as seen in the left column of the array.
%H A187537 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%F A187537 T(n,m) = m*Sum_{k=m..n} Sum_{i=k..n} binomial(i-1,k-1)*binomial(i,n-i)*Sum_{j=0..k} binomial(j,2*j-m-k)*binomial(k,j)/k, n>0, m<=n.
%e A187537      1,
%e A187537      3,    1,
%e A187537      9,    6,    1,
%e A187537     31,   27,    9,    1,
%e A187537    113,  116,   54,   12,   1,
%e A187537    431,  493,  282,   90,  15,   1,
%e A187537   1697, 2098, 1383,  556, 135,  18,  1,
%e A187537   6847, 8975, 6567, 3107, 965, 189, 21, 1
%o A187537 (Maxima)
%o A187537 T(n,m):=m*sum(sum(binomial(i-1,k-1)*binomial(i,n-i),i,k,n)*sum(binomial(j,2*j-m-k)*binomial(k,j),j,0,k)/k,k,m,n);
%K A187537 nonn,tabl
%O A187537 1,2
%A A187537 _Vladimir Kruchinin_, Mar 11 2011