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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.

%I A093905 #17 Jan 21 2017 09:23:02
%S A093905 1,1,3,1,5,11,1,7,26,50,1,9,47,154,274,1,11,74,342,1044,1764,1,13,107,
%T A093905 638,2754,8028,13068,1,15,146,1066,5944,24552,69264,109584,1,17,191,
%U A093905 1650,11274,60216,241128,663696,1026576,1,19,242,2414,19524,127860
%N 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.
%C A093905 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
%H A093905 G. C. Greubel, <a href="/A093905/b093905.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A093905 a(n, k) = (Product_{i=n-k..n} i)*(Sum_{i=n-k..n} 1/i), where a(n, 0) = 1.
%F A093905 a(n, k) = A067176(n, n-k-1) = A105954(k+1, n-k). Row sums are given by A093344.
%e A093905 Triangle begins:
%e A093905 1
%e A093905 1 3
%e A093905 1 5 11
%e A093905 1 7 26 50
%e A093905 1 9 47 154 274
%e A093905 ...
%e A093905 a(5, 3) = 4*3*2+5*3*2+5*4*2+5*4*3 = 154.
%t A093905 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 *)
%o A093905 (PARI) a(n, k) = prod(i=n-k, n, i)*sum(i=n-k,n,1/i);
%o A093905 tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(a(n,k), ", ")); print()); \\ _Michel Marcus_, Jan 21 2017
%Y A093905 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.
%Y A093905 Cf. A001705, A001711, A067176, A093344, A105954.
%Y A093905 A165674 is the reversal of this triangle. - _Johannes W. Meijer_, Oct 16 2009
%K A093905 nonn,easy,tabl
%O A093905 1,3
%A A093905 _Amarnath Murthy_, Apr 24 2004
%E A093905 Edited and extended by _David Wasserman_, Apr 24 2007