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

1,3

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

Links

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

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.

Example

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.

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 *)

Prog

(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

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

Sequence in context: A233037 A275999 A286910 * A324017 A063853 A219078

Adjacent sequences: A093902 A093903 A093904 * A093906 A093907 A093908

Keyword

nonn,easy,tabl

Author

Amarnath Murthy, Apr 24 2004

Extensions

Edited and extended by David Wasserman, Apr 24 2007

Status

approved