This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A185410 #47 Apr 01 2025 08:35:15
%S A185410 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,
%T A185410 1328,32,0,1,1086,21120,64240,42960,5664,64,0,1,3272,118632,660880,
%U A185410 900560,320064,23488,128,0,1,9832,638968,6049744,14713840,10725184,2225728,95872,256,0
%N A185410 A decomposition of the double factorials A001147.
%C A185410 Row sums are A001147. Reversal of A185411.
%C A185410 From _Peter Bala_, Jul 24 2012: (Start)
%C A185410 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.
%C A185410 Row reverse of A156919.
%C A185410 (End)
%C A185410 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
%H A185410 G. C. Greubel, <a href="/A185410/b185410.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185410 S.-M. Ma and T. Mansour, <a href="http://arxiv.org/abs/1409.6525">The 1/k-Eulerian polynomials and k-Stirling permutations</a>, arXiv preprint arXiv:1409.6525 [math.CO], 2014.
%H A185410 C. D. Savage and G. Viswanathan, <a href="https://doi.org/10.37236/16">The 1/k-Eulerian polynomials</a>, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012).
%F A185410 G.f.: 1/(1-x/(1-2xy/(1-3x/(1-4xy/(1-5x/(1-6xy/(1-7x/(1-8xy/(1- .... (continued fraction).
%F A185410 From _Peter Bala_, Jul 24 2012: (Start)
%F A185410 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.
%F A185410 Recurrence equation: T(n+1,k) = (2*k+1)*T(n,k) + 2*(n-k+1)*T(n,k-1).
%F A185410 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.
%F A185410 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.
%F A185410 Row sums A001147. Alt. row sums A202038. (End)
%F A185410 T(n,k) = 2^k*A102365(n,k). - _Philippe Deléham_, Feb 12 2013
%e A185410 Triangle begins:
%e A185410 1,
%e A185410 1, 0,
%e A185410 1, 2, 0,
%e A185410 1, 10, 4, 0,
%e A185410 1, 36, 60, 8, 0,
%e A185410 1, 116, 516, 296, 16, 0,
%e A185410 1, 358, 3508, 5168, 1328, 32, 0,
%e A185410 1, 1086, 21120, 64240, 42960, 5664, 64, 0,
%e A185410 1, 3272, 118632, 660880, 900560, 320064, 23488, 128, 0,
%e A185410 1, 9832, 638968, 6049744, 14713840, 10725184, 2225728, 95872, 256, 0,
%e A185410 ...
%e A185410 In the Savage-Viswanathan paper, the coefficients appear as
%e A185410 1
%e A185410 1 2
%e A185410 1 10 4
%e A185410 1 36 60 8
%e A185410 1 116 516 296 16
%e A185410 1 358 3508 5168 1328 32
%e A185410 1 1086 21120 64240 42960 5664 64
%e A185410 ...
%t A185410 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 *)
%Y A185410 Cf. A156919, A001147 (row sums), A112857, A173018, A186695, A202038 (alt. row sums).
%K A185410 nonn,easy,tabl
%O A185410 0,5
%A A185410 _Paul Barry_, Jan 26 2011