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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). 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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]

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]

LINKS

Table of n, a(n) for n=1..42.

EXAMPLE

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]

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 *)

CROSSREFS

Cf. A001700, A014449, A000312.

Row sums yield A000312 (Worpitzky's identity).

Cf. A008292.

Sequence in context: A128249 A071211 A222029 * A264902 A156653 A048159

Adjacent sequences:  A038672 A038673 A038674 * A038676 A038677 A038678

KEYWORD

nonn,tabl

AUTHOR

Alford Arnold

EXTENSIONS

More terms from Emeric Deutsch, May 08 2004

STATUS

approved

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Last modified October 21 13:54 EDT 2018. Contains 316423 sequences. (Running on oeis4.)