cp's OEIS Frontend

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

%I A156919 #75 Feb 16 2025 08:33:09
%S A156919 1,2,1,4,10,1,8,60,36,1,16,296,516,116,1,32,1328,5168,3508,358,1,64,
%T A156919 5664,42960,64240,21120,1086,1,128,23488,320064,900560,660880,118632,
%U A156919 3272,1,256,95872,2225728
%N A156919 Table of coefficients of polynomials related to the Dirichlet eta function.
%C A156919 Essentially the same as A185411. Row reverse of A185410. - _Peter Bala_, Jul 24 2012
%C A156919 The SF(z; n) formulas, see below, were discovered while studying certain properties of the Dirichlet eta function.
%C A156919 From _Peter Bala_, Apr 03 2011: (Start)
%C A156919 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,
%C A156919   D(g) = x*g^3
%C A156919   D^2(g) = x*(2 + x)*g^5
%C A156919   D^3(g) = x*(4 + 10*x + x^2)*g^7
%C A156919   D^4(g) = x*(8 + 60*x + 36*x^2 + x^3)*g^9.
%C A156919 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)
%H A156919 D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting Series Involving the Central Binomial Coefficient</a>, Am. Math. Monthly 92 (1985) 449-457, Polynomial V in eq (17). [R. J. Mathar, Feb 24 2009]
%H A156919 Shi-Mei Ma, <a href="http://arxiv.org/abs/1204.4963">A family of two-variable derivative polynomials for tangent and secant</a>, arXiv: 1204.4963v3 [math.CO], 2012.
%H A156919 Shi-Mei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p11">A family of two-variable derivative polynomials for tangent and secant</a>, El J. Combinat. 20 (1) (2013) P11
%H A156919 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 A156919 S.-M. Ma and Y.-N. Yeh, <a href="http://arxiv.org/abs/1503.06601">Stirling permutations, cycle structures of permutations and perfect matchings</a>, arXiv preprint arXiv:1503.06601 [math.CO], 2015.
%H A156919 Carla D. Savage and Gopal Viswanathan, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i1p9">The 1/k-Eulerian polynomials</a>, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012).
%H A156919 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet Eta Function</a>
%F A156919 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.
%F A156919 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.
%F A156919 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
%F A156919 From _Peter Bala_, Apr 03 2011 (Start)
%F A156919 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! + ... .
%F A156919 Explicit formula for the row polynomials:
%F A156919 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).
%F A156919 The polynomials x*(1 + x)^n * P(x/(x + 1),n) are the row polynomials of A187075.
%F A156919 The polynomials x^(n+1) * P((x + 1)/x,n) are the row polynomials of A186695.
%F A156919 Row sums are A001147(n+1). (End)
%F A156919 Sum_{k = 0..n} (-1)^k*T(n,k) = (-1)^binomial(n,2)*A012259(n+1). - _Johannes W. Meijer_, Sep 27 2011
%e A156919 The first few rows of the triangle are:
%e A156919   [1]
%e A156919   [2, 1]
%e A156919   [4, 10, 1]
%e A156919   [8, 60, 36, 1]
%e A156919   [16, 296, 516, 116, 1]
%e A156919 The first few P(z;n) are:
%e A156919   P(z; n=0) = 1
%e A156919   P(z; n=1) = 2 + z
%e A156919   P(z; n=2) = 4 + 10*z + z^2
%e A156919   P(z; n=3) = 8 + 60*z + 36*z^2 + z^3
%e A156919 The first few SF(z;n) are:
%e A156919   SF(z; n=0) = (1/2)*(1)/(1-z)^(3/2);
%e A156919   SF(z; n=1) = (1/4)*(2+z)/(1-z)^(5/2);
%e A156919   SF(z; n=2) = (1/8)*(4+10*z+z^2)/(1-z)^(7/2);
%e A156919   SF(z; n=3) = (1/16)*(8+60*z+36*z^2+z^3)/(1-z)^(9/2);
%e A156919 In the Savage-Viswanathan paper, the coefficients appear as
%e A156919   1;
%e A156919   1,    2;
%e A156919   1,   10,     4;
%e A156919   1,   36,    60,     8;
%e A156919   1,  116,   516,   296,    16;
%e A156919   1,  358,  3508,  5168,  1328,   32;
%e A156919   1, 1086, 21120, 64240, 42960, 5664, 64;
%e A156919   ...
%p A156919 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
%t A156919 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_ *)
%t A156919 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 *)
%Y A156919 A142963 and this sequence can be mapped onto the A156920 triangle.
%Y A156919 FP1 sequences A000340, A156922, A156923, A156924.
%Y A156919 FP2 sequences A050488, A142965, A142966, A142968.
%Y A156919 Appears in A162005, A000182, A162006 and A162007.
%Y A156919 Cf. A186695, A187075.
%Y A156919 Cf. A185410 (row reverse), A185411.
%K A156919 easy,nonn,tabl
%O A156919 0,2
%A A156919 _Johannes W. Meijer_, Feb 20 2009, Jun 24 2009
%E A156919 Minor edits from _Johannes W. Meijer_, Sep 27 2011