A049223 A convolution triangle of numbers obtained from A025750.

1, 10, 1, 150, 20, 1, 2625, 400, 30, 1, 49875, 8250, 750, 40, 1, 997500, 174750, 17875, 1200, 50, 1, 20662500, 3780000, 419625, 32500, 1750, 60, 1, 439078125, 83128125, 9810000, 839500, 53125, 2400, 70, 1, 9513359375, 1852500000, 229359375

Offset

1,2

Comments

a(n,1) = A025750(n); a(n,1)= 5^(n-1)*4*A034301(n-1)/n!, n >= 2. G.f. for m-th column: ((1-(1-25*x)^(1/5))/5)^m.

Links

Table of n, a(n) for n=1..39.

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

Formula

a(n, m) = 5*(5*(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.

T(n,m) = (m*sum(k=0..n-m, (-1)^(n-m-3*k)*binomial(n+k-1,n-1)*sum(j=0..k, 2^j*binomial(k,j)*sum(i=j..n-m-k+j, binomial(j,i-j)*binomial(k-j,n-m-3*(k-j)-i)*5^(3*(k-j)+i)))))/n. - Vladimir Kruchinin, Dec 10 2011

Prog

(Maxima)
T(n, m):=(m*sum((-1)^(n-m-3*k)*binomial(n+k-1, n-1)*sum(2^j*binomial(k, j)*sum(binomial(j, i-j)*binomial(k-j, n-m-3*(k-j)-i)*5^(3*(k-j)+i), i, j, n-m-k+j), j, 0, k), k, 0, n-m))/n; /* Vladimir Kruchinin, Dec 10 2011 */

Crossrefs

Cf. A048966, A049213. Row sums = A025758.

Sequence in context: A048882 A192357 A156286 * A308282 A223512 A131367

Adjacent sequences: A049220 A049221 A049222 * A049224 A049225 A049226

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Status

approved