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A038675 Triangle read by rows: T(n,k)=A(n,k)*binomial(n+k-1,n), where A(n,k) are the Eulerian numbers (A008292).

%I A038675 #17 Jun 14 2013 13:22:44
%S A038675 1,1,3,1,16,10,1,55,165,35,1,156,1386,1456,126,1,399,8456,25368,11970,
%T A038675 462,1,960,42876,289920,393030,95040,1716,1,2223,193185,2577135,
%U A038675 7731405,5525091,741741,6435,1,5020,803440,19411480,111675850,176644468
%N A038675 Triangle read by rows: T(n,k)=A(n,k)*binomial(n+k-1,n), where A(n,k) are the Eulerian numbers (A008292).
%C A038675 Andrews, Theory of Partitions, (1976), discussion of multisets.
%C A038675 Let a = a_1,a_2,...,a_n be a sequence on the alphabet {1,2,...,n}. Scan a from left to right and create an n-permutation by noting the POSITION of the elements as you come to them in order from least to greatest. See example. T(n,k) is the number of sequences that correspond to such a permutation having exactly n-k descents. [From _Geoffrey Critzer_, May 19 2010]
%D A038675 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, Reading, Mass., 1994, p. 269 (Worpitzky's identity).
%D A038675 Miklos Bona, Combinatorics of Permutations,Chapman and Hall,2004,page 6. [From _Geoffrey Critzer_, May 19 2010]
%e A038675 1;
%e A038675 1,3;
%e A038675 1,16,10;
%e A038675 1,55,165,35;
%e A038675 1,156,1386,1456,126;
%e A038675 ...
%e A038675 If a = 3,1,1,2,4,3 the corresponding 6-permutation is 2,3,4,1,6,5 because the first 1 is in the 2nd position, the second 1 is in the 3rd position,the 2 is in the 4th position, the first 3 is in the first position, the next 3 is in the 6th position and the 4 is in the 5th position of the sequence a. [From _Geoffrey Critzer_, May 19 2010]
%p A038675 A:=(n,k)->sum((-1)^j*(k-j)^n*binomial(n+1,j),j=0..k): T:=(n,k)->A(n,k)*binomial(n+k-1,n): seq(seq(T(n,k),k=1..n),n=1..10);
%t A038675 Table[Table[Eulerian[n, k] Binomial[n + k, n], {k, 0, n - 1}], {n, 1,10}] (* _Geoffrey Critzer_, Jun 13 2013 *)
%Y A038675 Cf. A001700, A014449, A000312.
%Y A038675 Row sums yield A000312 (Worpitzky's identity).
%Y A038675 Cf. A008292.
%K A038675 nonn,tabl
%O A038675 1,3
%A A038675 _Alford Arnold_
%E A038675 More terms from _Emeric Deutsch_, May 08 2004