A185410 -id:A185410 - cp's OEIS frontend

A156919 Table of coefficients of polynomials related to the Dirichlet eta function.

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

Offset: 0

Johannes W. Meijer, Feb 20 2009, Jun 24 2009

Essentially the same as A185411. Row reverse of A185410. - Peter Bala, Jul 24 2012
The SF(z; n) formulas, see below, were discovered while studying certain properties of the Dirichlet eta function.
From Peter Bala, Apr 03 2011: (Start)
Let D be the differential operator 2*x*d/dx. The row polynomials of this table come from repeated application of the operator D to the function g(x) = 1/sqrt(1 - x). For example,
  D(g) = x*g^3
  D^2(g) = x*(2 + x)*g^5
  D^3(g) = x*(4 + 10*x + x^2)*g^7
  D^4(g) = x*(8 + 60*x + 36*x^2 + x^3)*g^9.
Thus this triangle is analogous to the triangle of Eulerian numbers A008292, whose row polynomials come from the  repeated application of the operator x*d/dx to the function 1/(1 - x). (End)

			The first few rows of the triangle are:
  [1]
  [2, 1]
  [4, 10, 1]
  [8, 60, 36, 1]
  [16, 296, 516, 116, 1]
The first few P(z;n) are:
  P(z; n=0) = 1
  P(z; n=1) = 2 + z
  P(z; n=2) = 4 + 10*z + z^2
  P(z; n=3) = 8 + 60*z + 36*z^2 + z^3
The first few SF(z;n) are:
  SF(z; n=0) = (1/2)*(1)/(1-z)^(3/2);
  SF(z; n=1) = (1/4)*(2+z)/(1-z)^(5/2);
  SF(z; n=2) = (1/8)*(4+10*z+z^2)/(1-z)^(7/2);
  SF(z; n=3) = (1/16)*(8+60*z+36*z^2+z^3)/(1-z)^(9/2);
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;
  ...

D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449-457, Polynomial V in eq (17). [R. J. Mathar, Feb 24 2009]
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv: 1204.4963v3 [math.CO], 2012.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11
S.-M. Ma and T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint arXiv:1409.6525 [math.CO], 2014.
S.-M. Ma and Y.-N. Yeh, Stirling permutations, cycle structures of permutations and perfect matchings, arXiv preprint arXiv:1503.06601 [math.CO], 2015.
Carla D. Savage and Gopal Viswanathan, The 1/k-Eulerian polynomials, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012).
Eric Weisstein's World of Mathematics, Dirichlet Eta Function

A142963 and this sequence can be mapped onto the A156920 triangle.
FP1 sequences A000340, A156922, A156923, A156924.
FP2 sequences A050488, A142965, A142966, A142968.
Appears in A162005, A000182, A162006 and A162007.
Cf. A186695, A187075.
Cf. A185410 (row reverse), A185411.

A156919 := proc(n,m) if n=m then 1; elif m=0 then 2^n ; elif m<0 or m>n then 0; else 2*(m+1)*procname(n-1,m)+(2*n-2*m+1)*procname(n-1,m-1) ; end if; end proc: seq(seq(A156919(n,m), m=0..n), n=0..7); # R. J. Mathar, Feb 03 2011

g[0] = 1/Sqrt[1-x]; g[n_] := g[n] = 2x*D[g[n-1], x]; p[n_] := g[n] / g[0]^(2n+1) // Cancel; row[n_] := CoefficientList[p[n], x] // Rest; Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Aug 09 2012, after Peter Bala *)
Flatten[Table[Rest[CoefficientList[Nest[2 x D[#, x] &, (1 - x)^(-1/2), k] (1 - x)^(k + 1/2), x]], {k, 10}]] (* Jan Mangaldan, Mar 15 2013 *)

SF(z; n) = Sum_{m >= 1} m^(n-1)*4^(-m)*z^(m-1)*Gamma(2*m+1)/(Gamma(m)^2) = P(z;n) / (2^(n+1)*(1-z)^((2*n+3)/2)) for n >= 0. The polynomials P(z;n) = Sum_{k = 0..n} a(k)*z^k generate the a(n) sequence.
If we write the sequence as a triangle the following relation holds: T(n,m) = (2*m+2)*T(n-1,m) + (2*n-2*m+1)*T(n-1,m-1) with T(n,m=0) = 2^n and T(n,n) = 1, n >= 0 and 0 <= m <= n.
G.f.: 1/(1-xy-2x/(1-3xy/(1-4x/(1-5xy/(1-6x/(1-7xy/(1-8x/(1-... (continued fraction). - Paul Barry, Jan 26 2011
From Peter Bala, Apr 03 2011 (Start)
E.g.f.: exp(z*(x + 2)) * (1 - x)/(exp(2*x*z) - x*exp(2*z))^(3/2) = Sum_{n >= 0} P(x,n)*z^n/n! = 1 + (2 + x)*z + (4 + 10*x + x^2)*z^2/2! + (8 + 60*x + 36*x^2 + x^3)*z^3/3! + ... .
Explicit formula for the row polynomials:
P(x,n-1) = Sum_{k = 1..n} 2^(n-2*k)*binomial(2k,k)*k!*Stirling2(n,k)*x^(k-1)*(1 - x)^(n-k).
The polynomials x*(1 + x)^n * P(x/(x + 1),n) are the row polynomials of A187075.
The polynomials x^(n+1) * P((x + 1)/x,n) are the row polynomials of A186695.
Row sums are A001147(n+1). (End)
Sum_{k = 0..n} (-1)^k*T(n,k) = (-1)^binomial(n,2)*A012259(n+1). - Johannes W. Meijer, Sep 27 2011

Minor edits from Johannes W. Meijer, Sep 27 2011

A185411 A triangular decomposition of the double factorial numbers A001147.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jan 26 2011

Row sums are A001147. Reversal of A185410. Contains A156919 as submatrix.
Row n counts perfect matchings of [2n] by number of matches in which the smaller entry is odd. For example, T(2,1)=2 counts 13/24, 14/23, in each of which only the first matching pair has an odd smaller entry. Outline proof. Consider the map on perfect matchings of [2n] given by "delete the entries n and n-1 and, if they were not originally matched to each other, match up their now-unmatched partners". Consideration of this map and its effect on the statistic "number of matches in which the smaller entry is odd" yields the Mathematica recurrence below. - David Callan, Dec 13 2011
Triangle T(n,k), 0 <= k <= n, given by (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 11, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 12 2013
T(n,k), 0 <= k <= n, is the number of signed permutations of [n] that are products of balanced cycles (i.e., cuspidal elements of the type B Coxeter group) and have excedance number of type B equal to k. - Jose Bastidas, Jul 05 2023

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     1;
  0,   4,    10,       1;
  0,   8,    60,      36,        1;
  0,  16,   296,     516,      116,        1;
  0,  32,  1328,    5168,     3508,      358,       1;
  0,  64,  5664,   42960,    64240,    21120,    1086,      1;
  0, 128, 23488,  320064,   900560,   660880,  118632,   3272,    1;
  0, 256, 95872, 2225728, 10725184, 14713840, 6049744, 638968, 9832, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Jose Bastidas, Christophe Hohlweg, and Franco Saliola, The Primitive Eulerian polynomial, arXiv:2306.15556 [math.CO], 2023. See Table 2 p. 18.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv: 1204.4963v3 [math.CO], 2012.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, Elect. J. Combinat. 20 (1) (2013) #P11.
Shi-Mei Ma, T. Mansour, and D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.
Shi-Mei Ma, T. Mansour and D. G. L. Wang, Combinatorics of Dumont differential system on the Jacobi elliptic functions, arXiv:1403.0233 [math.CO], 2014.
Shi-Mei Ma, Toufik Mansour, David G.L. Wang, and Yeong-Nan Yeh, Several variants of the Dumont differential system and permutation statistics, Science China Mathematics 60 (2018).
Shi-Mei Ma and Y.-N. Yeh, Stirling permutations, cycle structures of permutations and perfect matchings, arXiv:1503.06601 [math.CO], 2015.

Columns 0-1 give: A000007, A131577.

Cf. A001147, A185410, A156919 (another version).

u[n_, 0] := If[n==0, 1, 0]; u[n_, m_] /; m==1 := 2^(n - 1); u[n_, m_] /; m==n>=1 := 1; u[n_, m_] /; 1David Callan, Dec 13 2011 *)

G.f.: 1/(1-xy/(1-2x/(1-3xy/(1-4x/(1-5xy/(1-6x/(1-7xy/(1- ... (continued fraction).
T(n,k) = (2n-2k+1)*T(n-1,k-1) + 2k*T(n-1,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n. - Philippe Deléham, Feb 12 2013
T(n,k) = 2^(n-k)*A211399(n,k). - Philippe Deléham, Feb 12 2013

Sequence terms corrected by Paul Barry, Jan 27 2011

A256978 Irregular triangle read by rows: coefficients of polynomials related to Stirling permutations.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 3, 1, 1, 7, 29, 31, 29, 7, 1, 1, 15, 101, 195, 321, 195, 101, 15, 1, 1, 31, 327, 1001, 2507, 2661, 2507, 1001, 327, 31, 1, 1, 63, 1023, 4641, 16479, 26481, 37759, 26481, 16479, 4641, 1023, 63, 1, 1, 127, 3145, 20343, 98289, 221775, 439105, 461455, 439105, 221775, 98289, 20343, 3145, 127, 1

Offset: 1

N. J. A. Sloane, Apr 23 2015

			Triangle begins:
  n\k | 1   2   3    4    5    6    7    8   9  10  11
  ----+-----------------------------------------------
    1 | 1
    2 | 1   1   1
    3 | 1   3   7    3    1
    4 | 1   7  29   31   29    7    1
    5 | 1  15 101  195  321  195  101   15   1
    6 | 1  31 327 1001 2507 2661 2507 1001 327  31   1
    ...

Shi-Mei Ma and Toufik Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint, arXiv:1409.6525 [math.CO], 2014. See polynomials C_n(x).
Shi-Mei Ma and Toufik Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, Discrete Mathematics, Volume 338, Issue 8, 2015, 1468-1472.
Shi-Mei Ma, Yeong-Nan Yeh, The alternating run polynomials of permutations, arXiv:1904.11437 [math.CO], 2019. See p. 9.

Cf. A185410.

gf : taylor((exp(z*(x - 1)*(x + 1)) + x)/(x + 1)*sqrt((1 - x^2)/(exp(2*z*(x - 1)*(x + 1)) - x^2)) - 1, z, 0, 50)$
row(x, n) := n!*ratcoef(gf, z, n)$
create_list(ratcoef(row(x, n), x, k), n, 1, 20, k, 1, hipow(row(x, n), x));
/* Franck Maminirina Ramaharo, Feb 05 2019 */

E.g.f.: (exp(z*(x - 1)*(x + 1)) + x)/(x + 1)*sqrt((1 - x^2)/(exp(2*z*(x - 1)*(x + 1)) - x^2)) - 1. - Franck Maminirina Ramaharo, Feb 05 2019

More terms from Franck Maminirina Ramaharo, Feb 05 2019

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A156919 Table of coefficients of polynomials related to the Dirichlet eta function.

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A185411 A triangular decomposition of the double factorial numbers A001147.

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A256978 Irregular triangle read by rows: coefficients of polynomials related to Stirling permutations.

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