A014449 -id:A014449 - cp's OEIS frontend

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

Alford Arnold

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]

			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]

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]

Cf. A001700, A014449, A000312.
Row sums yield A000312 (Worpitzky's identity).
Cf. A008292.

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);

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

More terms from Emeric Deutsch, May 08 2004

A014732 Squares of numbers in triangle of Eulerian numbers that are not 1.

Original entry on oeis.org

16, 121, 121, 676, 4356, 676, 3249, 91204, 91204, 3249, 14400, 1418481, 5837056, 1418481, 14400, 61009, 18429849, 243953161, 243953161, 18429849, 61009, 252004, 213393664, 7785238756, 24395316100, 7785238756, 213393664, 252004

Offset: 1

Mohammad K. Azarian

Cf. A008292, A014449.

a(n) = A014449(n)^2. - Sean A. Irvine, Nov 18 2018

More terms from James Sellers

Offset corrected by Joerg Arndt, Nov 19 2018

cp's OEIS Frontend

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).

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