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 A156919 Table of coefficients of polynomials related to the Dirichlet eta function. 21

%I

%S 1,2,1,4,10,1,8,60,36,1,16,296,516,116,1,32,1328,5168,3508,358,1,64,

%T 5664,42960,64240,21120,1086,1,128,23488,320064,900560,660880,118632,

%U 3272,1,256,95872,2225728

%N Table of coefficients of polynomials related to the Dirichlet eta function.

%C Essentially the same as A185411. Row reverse of A185410. - _Peter Bala_, Jul 24 2012

%C The SF(z; n) formulas, see below, were discovered while studying certain properties of the Dirichlet eta function.

%C From _Peter Bala_, Apr 03 2011 (Start)

%C 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 D(g) = x*g^3

%C D^2(g) = x*(2+x)*g^5

%C D^3(g) = x*(4+10*x+x^2)*g^7

%C D^4(g) = x*(8+60*x+36*x^2+x^3)*g^9.

%C 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 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 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 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 S.-M. Ma, 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 S.-M. Ma, 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet Eta Function</a>

%F SF(z; n) = sum(m^(n-1)*4^(-m)*z^(m-1)*GAMMA(2*m+1)/(GAMMA(m)^2), m=1..infinity) = P(z;n) / (2^(n+1)*(1-z)^((2*n+3)/2)) for n >= 0. The polynomials P(z;n) = sum( a(k)*z^k, k=0..n) generate the a(n) sequence.

%F 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 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 From _Peter Bala_, Apr 03 2011 (Start)

%F E.g.f.: exp(z*(x+2)){(1-x)/(exp(2*x*z)-x*exp(2*z))}^(3/2) = sum {n = 0..inf} 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 Explicit formula for the row polynomials:

%F 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 The polynomials x*(1+x)^n*P(x/(x+1),n) are the row polynomials of A187075.

%F The polynomials x^(n+1)*P((x+1)/x,n) are the row polynomials of A186695.

%F Row sums are A001147(n+1). (End)

%F sum((-1)^k*T(n,k), k=0..n) = (-1)^binomial(n,2)*A012259(n+1) [_Johannes W. Meijer_, Sep 27, 2011]

%e The first few rows of the triangle are:

%e [1]

%e [2, 1]

%e [4, 10, 1]

%e [8, 60, 36, 1]

%e [16, 296, 516, 116, 1]

%e The first few P(z;n) are:

%e P(z; n=0) = 1

%e P(z; n=1) = 2 + z

%e P(z; n=2) = 4 + 10*z + z^2

%e P(z; n=3) = 8 + 60*z + 36*z^2 + z^3

%e The first few SF(z;n) are:

%e SF(z; n=0) = (1/2)*(1)/(1-z)^(3/2);

%e SF(z; n=1) = (1/4)*(2+z)/(1-z)^(5/2);

%e SF(z; n=2) = (1/8)*(4+10*z+z^2)/(1-z)^(7/2);

%e SF(z; n=3) = (1/16)*(8+60*z+36*z^2+z^3)/(1-z)^(9/2);

%e In the Savage-Viswanathan paper, the coefficients appear as

%e 1

%e 1 2

%e 1 10 4

%e 1 36 60 8

%e 1 116 516 296 16

%e 1 358 3508 5168 1328 32

%e 1 1086 21120 64240 42960 5664 64

%e ...

%p 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 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 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 A142963 and this sequence can be mapped onto the A156920 triangle.

%Y FP1 sequences A000340, A156922, A156923, A156924.

%Y FP2 sequences A050488, A142965, A142966, A142968.

%Y Appears in A162005, A000182, A162006 and A162007.

%Y Cf. A186695, A187075. [From _Peter Bala_, Apr 03 2011]

%Y Cf. A185410 (row reverse), A185411.

%K easy,nonn,tabl

%O 0,2

%A _Johannes W. Meijer_, Feb 20 2009, Jun 24 2009

%E Minor edits from _Johannes W. Meijer_, Sep 27 2011

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Last modified January 18 23:05 EST 2019. Contains 319282 sequences. (Running on oeis4.)