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A201637
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Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
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22
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1, 1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 22, 58, 24, 0, 1, 52, 328, 444, 120, 0, 1, 114, 1452, 4400, 3708, 720, 0, 1, 240, 5610, 32120, 58140, 33984, 5040, 0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320, 0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880, 0
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table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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This version indexes the Eulerian numbers in the same way as Graham et al.'s Concrete Mathematics. This indexing is also used by Maple. The indexing as used by Riordan, Comtet and others, is given in A008517, which is the main entry for the second-order Eulerian numbers.
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 256.
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LINKS
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EXAMPLE
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... [0] [1] [2] [3] [4] [5] [6] [7] [8]
[0] [1]
[1] [1, 0]
[2] [1, 2, 0]
[3] [1, 8, 6, 0]
[4] [1, 22, 58, 24, 0]
[5] [1, 52, 328, 444, 120, 0]
[6] [1, 114, 1452, 4400, 3708, 720, 0]
[7] [1, 240, 5610, 32120, 58140, 33984, 5040, 0]
[8] [1, 494, 19950, 195800, 644020, 785304, 341136, 40320, 0]
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MAPLE
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A201637 := (n, k) -> combinat[eulerian2](n, k):
for n from 0 to 9 do seq(A201637(n, k), k=0..n) od;
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MATHEMATICA
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t[0, 0] = 1; t[n_, m_] = Sum[(-1)^(n+k)*Binomial[2*n+1, k]*StirlingS1[2*n-m-k, n-m-k], {k, 0, n-m-1}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten
E2[n_, k_] /; k == 0 = 1; E2[n_, k_] /; k < 0 || k > n = 0;
E2[n_, k_] := E2[n, k] = (2*n - 1 - k)*E2[n-1, k-1] + (k + 1)*E2[n-1, k];
Table[E2[n, k], {n, 0, 8}, {k, 0, n}] // TableForm
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PROG
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(Sage)
@CachedFunction
def eulerian2(n, k):
if k==0: return 1
if k==n: return 0
return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
for n in (0..9): [eulerian2(n, k) for k in(0..n)]
(PARI) for(n=0, 10, for(m=0, n, print1(if(m==0 || n==0, 1, sum(k=0, n-m-1, (-1)^(n+k)* binomial(2*n+1, k)*stirling(2*n-m-k, n-m-k, 1))), ", "))) \\ G. C. Greubel, Oct 24 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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