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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. 6

%I

%S 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 638,2754,8028,13068,1,15,146,1066,5944,24552,69264,109584,1,17,191,

%U 1650,11274,60216,241128,663696,1026576,1,19,242,2414,19524,127860

%N 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 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 G. C. Greubel, <a href="/A093905/b093905.txt">Table of n, a(n) for the first 50 rows, flattened</a>

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

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

%e Triangle begins:

%e 1

%e 1 3

%e 1 5 11

%e 1 7 26 50

%e 1 9 47 154 274

%e ...

%e a(5, 3) = 4*3*2+5*3*2+5*4*2+5*4*3 = 154.

%t 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 (PARI) a(n, k) = prod(i=n-k, n, i)*sum(i=n-k,n,1/i);

%o tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(a(n,k), ", ")); print()); \\ _Michel Marcus_, Jan 21 2017

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

%Y A165674 is the reversal of this triangle. - _Johannes W. Meijer_, Oct 16 2009

%K nonn,easy,tabl

%O 1,3

%A _Amarnath Murthy_, Apr 24 2004

%E Edited and extended by _David Wasserman_, Apr 24 2007

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Last modified January 21 21:30 EST 2020. Contains 331128 sequences. (Running on oeis4.)