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

1,2

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.

Links

Table of n, a(n) for n=1..45.

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

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.

Example

     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

Prog

(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);

Crossrefs

Sequence in context: A027465 A164942 A236420 * A246256 A157393 A242402

Adjacent sequences: A187534 A187535 A187536 * A187538 A187539 A187540

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Mar 11 2011

Status

approved