A049213 A convolution triangle of numbers obtained from A025749.

1, 6, 1, 56, 12, 1, 616, 148, 18, 1, 7392, 1904, 276, 24, 1, 93632, 25312, 4080, 440, 30, 1, 1230592, 344960, 59808, 7360, 640, 36, 1, 16612992, 4792128, 876960, 118224, 11960, 876, 42, 1, 228890112, 67586816, 12900416, 1860992, 209200, 18096, 1148

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• tabl

Comments

a(n,1) = A025749(n); a(n,1)= 4^(n-1)*3*A034176(n-1)/n!, n >= 2.

G.f. for m-th column: ((1-(1-16*x)^(1/4))/4)^m.

Links

• W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Crossrefs

Cf. A048966. Row sums = A025757.

Programs

Mathematica

a[n_, n_] = 1; a[n_, m_] := m/n * 4^(n-m) * Sum[ Binomial[n+k-1, n-1] * Sum[ Binomial[j, n-m-3*k+2*j] * 4^(j-k) * Binomial[k, j] * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j), {j, 0, k}], {k, 1, n-m}]; Table[a[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)

Formula

a(n, m) = 4*(4*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0; a(1, 1)=1.

a(n,m) = (m/n) * 4^(n-m) * Sum_{k=1..n-m} binomial(n+k-1, n-1) * Sum_{j=0..k} binomial(j, n-m-3*k+2*j) * 4^(j-k) * binomial(k,j) * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j), n > m; a(n,n)=1. - Vladimir Kruchinin, Feb 08 2011