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

1, 3, 1, 9, 6, 1, 31, 27, 9, 1, 113, 116, 54, 12, 1, 431, 493, 282, 90, 15, 1, 1697, 2098, 1383, 556, 135, 18, 1, 6847, 8975, 6567, 3107, 965, 189, 21, 1, 28161, 38640, 30636, 16376, 6070, 1536, 252, 24, 1

Offset: 1

Author

Vladimir Kruchinin, Mar 11 2011

Keywords

• nonn
• tabl

Comments

The column with index 0 of the standard array is not incorporated in this triangle. (It contains a 1 followed by zeros.)

The truncated Fibonacci sequence is A000045(x)/x-1 = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5+ ...

The composition with the Motzkin sequence is A001006(...) = 1 + x + 4*x^2 + 15*x^3 + 58*x^4 + 229*x^5 + ...

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.

Examples

			     1,
     3,    1,
     9,    6,    1,
    31,   27,    9,    1,
   113,  116,   54,   12,   1,
   431,  493,  282,   90,  15,   1,
  1697, 2098, 1383,  556, 135,  18,  1,
  6847, 8975, 6567, 3107, 965, 189, 21, 1
		

Links

• Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Programs

Maxima

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);

Formula

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.