A185815 Exponential Riordan array (log(1/(1-x)), x*A005043(x)).

0, 1, 0, 1, 2, 0, 2, 3, 3, 0, 6, 32, 6, 4, 0, 24, 210, 140, 10, 5, 0, 120, 2904, 1170, 400, 15, 6, 0, 720, 41580, 22344, 3990, 910, 21, 7, 0, 5040, 789984, 379680, 98784, 10500, 1792, 28, 8, 0, 40320, 16961616, 8595936, 1930320, 325584, 23436, 3192, 36, 9, 0

Offset: 0

Author

Vladimir Kruchinin, Feb 05 2011

Keywords

• nonn
• tabl

Examples

			Array begins:
    0;
    1,     0;
    1,     2,     0;
    2,     3,     3,    0;
    6,    32,     6,    4,   0;
   24,   210,   140,   10,   5,  0;
  120,  2904,  1170,  400,  15,  6, 0;
  720, 41580, 22344, 3990, 910, 21, 7, 0;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Programs

Maple

A185815 := proc(n,k) if n = k then 0; elif k = 0 then (n-1)! ; else n!/(k-1)!*add(1/i/(n-i)*add(binomial(2*j-k-1,j-1)*(-1)^(n-j-i)*binomial(n-i,j),j=k..n-i),i=1..n-k) ; end if; end proc:
seq(seq(A185815(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Feb 09 2011

Mathematica

t[n_, k_] := n!/(k-1)!*Sum[ 1/(i*(n-i))*((-1)^(n+k-i)*(n-i)!*HypergeometricPFQ[ {(k+1)/2, k/2, i+k-n}, {k, k+1}, 4]) / (k!*(n-k-i)!), {i, 1, n-k}]; t[0, 0] = 0; t[n_, 0] := (n-1)!; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 01 2013, after given formula *)

Formula

R(n,k):= (n!/(k-1)!)*Sum_{i=1..(n-k)} (1/(i*(n-i)))*Sum_{j=k..(n-i)} binomial(2*j-k-1,j-1)*(-1)^(n-j-i)*binomial(n-i,j), k>0, R(0,0)=0, R(n,0)=(n-1)!.

Extensions

More terms from Michel Marcus, Feb 19 2025