A185663 Riordan array (A000045(x)^m, x*A000108(x)), m = 2.

0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 5, 3, 1, 0, 0, 10, 9, 4, 1, 0, 0, 20, 24, 14, 5, 1, 0, 0, 38, 64, 44, 20, 6, 1, 0, 0, 71, 173, 135, 71, 27, 7, 1, 0, 0, 130, 485, 414, 241, 106, 35, 8, 1, 0, 0, 235, 1420, 1290, 805, 391, 150, 44, 9, 1, 0, 0

Offset

0,7

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2013.

Formula

R(n,k,m) = k*Sum_{i=0..n-k} (Sum_{j=ceiling((i-m)/2)..i-m} binomial(j, i-m-j) * binomial(m+j-1, m-1)) * binomial(2*(n-i)-k-1, n-i-1)/(n-i), k > 0.

R(n,0,m) = Sum_{j=ceiling((n-m)/2)..n-m} binomial(j,n-m-j) * binomial(m+j-1,m-1), m = 2.

Example

Array begins
   0;
   0,  0;
   1,  0,  0;
   2,  1,  0,  0;
   5,  3,  1,  0,  0;
  10,  9,  4,  1,  0,  0;
  20, 24, 14,  5,  1,  0,  0;
  38, 64, 44, 20,  6,  1,  0,  0;

Mathematica

R[n_, k_, m_] := k*Sum[Sum[Binomial[j, i - m - j]*Binomial[m - 1 + j, m - 1], {j, Ceiling[(i - m)/2], i - m}]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k} ]; R[n_, 0, m_] := Sum[Binomial[j, n - m - j]*Binomial[j + m - 1, m - 1], {j, Ceiling[(n - m)/2], n - m}];
Table[R[n, k, 2], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Jul 23 2017 *)

Crossrefs

Cf. A185937, A139375 (m = 1).

Sequence in context: A246187 A079508 A057150 * A262125 A360068 A105868

Adjacent sequences: A185660 A185661 A185662 * A185664 A185665 A185666

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Feb 08 2011

Status

approved