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).

1, 1, 3, 1, 16, 10, 1, 55, 165, 35, 1, 156, 1386, 1456, 126, 1, 399, 8456, 25368, 11970, 462, 1, 960, 42876, 289920, 393030, 95040, 1716, 1, 2223, 193185, 2577135, 7731405, 5525091, 741741, 6435, 1, 5020, 803440, 19411480, 111675850, 176644468

Offset: 1

Author

Alford Arnold

Keywords

• nonn
• tabl

Comments

Andrews, Theory of Partitions, (1976), discussion of multisets.

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]

Examples

			1;
1,3;
1,16,10;
1,55,165,35;
1,156,1386,1456,126;
...
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]
		

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, Reading, Mass., 1994, p. 269 (Worpitzky's identity).
• Miklos Bona, Combinatorics of Permutations,Chapman and Hall,2004,page 6. [From Geoffrey Critzer, May 19 2010]

Crossrefs

Cf. A001700, A014449, A000312.

Row sums yield A000312 (Worpitzky's identity).

Cf. A008292.

Programs

Maple

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);

Mathematica

Table[Table[Eulerian[n, k] Binomial[n + k, n], {k, 0, n - 1}], {n, 1,10}] (* Geoffrey Critzer, Jun 13 2013 *)

Extensions

More terms from Emeric Deutsch, May 08 2004