A093905 Triangle read by rows: for 0 <= k < n, a(n, k) is the sum of the products of all subsets of {n-k, n-k+1, ..., n} with k members.

1, 1, 3, 1, 5, 11, 1, 7, 26, 50, 1, 9, 47, 154, 274, 1, 11, 74, 342, 1044, 1764, 1, 13, 107, 638, 2754, 8028, 13068, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576, 1, 19, 242, 2414, 19524, 127860

Offset: 1

Author

Amarnath Murthy, Apr 24 2004

Keywords

• nonn
• easy
• tabl

Comments

Triangle A165674, which is the reversal of this triangle, is generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n). - Johannes W. Meijer, Oct 16 2009

Examples

			Triangle begins:
1
1 3
1 5 11
1 7 26 50
1 9 47 154 274
...
a(5, 3) = 4*3*2+5*3*2+5*4*2+5*4*3 = 154.
		

Links

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

Crossrefs

The leading diagonal is given by A000254, Stirling numbers of first kind. The next nine diagonals are A001705, A001711, A001716, A001721, A051524, A051545, A051560, A051562 and A051564, generalized Stirling numbers.

Cf. A001705, A001711, A067176, A093344, A105954.

A165674 is the reversal of this triangle. - Johannes W. Meijer, Oct 16 2009

Programs

Mathematica

T[n_, 0] := 1; T[n_, k_]:= Product[i, {i, n - k, n}]*Sum[1/i, {i, n - k, n}]; Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] (* G. C. Greubel, Jan 21 2017 *)

PARI

a(n, k) = prod(i=n-k, n, i)*sum(i=n-k,n,1/i);
tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(a(n,k), ", ")); print()); \\ Michel Marcus, Jan 21 2017

Formula

a(n, k) = (Product_{i=n-k..n} i)*(Sum_{i=n-k..n} 1/i), where a(n, 0) = 1.

a(n, k) = A067176(n, n-k-1) = A105954(k+1, n-k). Row sums are given by A093344.

Extensions

Edited and extended by David Wasserman, Apr 24 2007