A185410 A decomposition of the double factorials A001147.

1, 1, 0, 1, 2, 0, 1, 10, 4, 0, 1, 36, 60, 8, 0, 1, 116, 516, 296, 16, 0, 1, 358, 3508, 5168, 1328, 32, 0, 1, 1086, 21120, 64240, 42960, 5664, 64, 0, 1, 3272, 118632, 660880, 900560, 320064, 23488, 128, 0, 1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0

Offset: 0

Author

Paul Barry, Jan 26 2011

Keywords

• nonn
• easy
• tabl

Comments

Row sums are A001147. Reversal of A185411.

From Peter Bala, Jul 24 2012: (Start)

This is the case k = 2 of the 1/k—Eulerian polynomials introduced by Savage and Viswanathan. They give a combinatorial interpretation of the triangle in terms of an ascent statistic on sets of inversion sequences and a geometric interpretation in terms of lecture hall polytopes.

Row reverse of A156919.

(End)

Triangle T(n,k), 0<=k<=n, given by (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) DELTA (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 12 2013

Examples

			Triangle begins:
  1,
  1,    0,
  1,    2,      0,
  1,   10,      4,       0,
  1,   36,     60,       8,        0,
  1,  116,    516,     296,       16,        0,
  1,  358,   3508,    5168,     1328,       32,       0,
  1, 1086,  21120,   64240,    42960,     5664,      64,     0,
  1, 3272, 118632,  660880,   900560,   320064,   23488,   128,   0,
  1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0,
  ...
In the Savage-Viswanathan paper, the coefficients appear as
  1
  1    2
  1   10     4
  1   36    60     8
  1  116   516   296    16
  1  358  3508  5168  1328   32
  1 1086 21120 64240 42960 5664 64
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• S.-M. Ma and T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint arXiv:1409.6525 [math.CO], 2014.
• C. D. Savage and G. Viswanathan, The 1/k-Eulerian polynomials, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012).

Crossrefs

Cf. A156919, A001147 (row sums), A112857, A173018, A186695, A202038 (alt. row sums).

Programs

Mathematica

T[0, 0] := 1;  T[n_, -1] := 0;  T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, 2^k*T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* G. C. Greubel, Jun 30 2017 *)

Formula

G.f.: 1/(1-x/(1-2xy/(1-3x/(1-4xy/(1-5x/(1-6xy/(1-7x/(1-8xy/(1- .... (continued fraction).

From Peter Bala, Jul 24 2012: (Start)

T(n,k) = sum {j=0..k}(-1)^(k-j)/4^j*C(n+1/2,k-j)*C(2*j,j)*(2*j+1)^n.

Recurrence equation: T(n+1,k) = (2*k+1)*T(n,k) + 2*(n-k+1)*T(n,k-1).

E.g.f.: sqrt(E(x,2*z)) = 1 + z + (1+2*x)*z^2/2! + (1+10*x+4*x^2)*z^3/3! + ..., where E(x,z) = (1-x)/(exp(z*(x-1)) - x) is the e.g.f. for the Eulerian numbers (version A173018). Cf. A156919.

Row polynomial R(n,x) = sum {k = 1..n} 2^(n-2*k)*C(2*k,k)*k!*Stirling2(n,k)*(x-1)^(n-k). R(n,4*x)/(1-4*x)^(n+1/2) = sum {k>=0} C(2*k,k)*(2*k+1)^n*x^k. The sequence of rational functions x*R(n,x)/(1-x)^(n+1) conjecturally occurs in the first column of (I - x*A112857)^(-1). (1+x)^(n-1)*R(n,x/(x+1)) gives the n-th row polynomial of A186695.

Row sums A001147. Alt. row sums A202038. (End)

T(n,k) = 2^k*A102365(n,k). - Philippe Deléham, Feb 12 2013