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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 A350446 A156653 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 September 28 07:22 EDT 2023. Contains 365724 sequences. (Running on oeis4.)