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 A156919 Table of coefficients of polynomials related to the Dirichlet eta function. 21
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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) LINKS 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, T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint arXiv:1409.6525 [math.CO], 2014. S.-M. Ma, 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 FORMULA 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. 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..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! +... 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((-1)^k*T(n,k), k=0..n) = (-1)^binomial(n,2)*A012259(n+1) [Johannes W. Meijer, Sep 27, 2011] EXAMPLE The first few rows of the triangle are:      [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   ... MAPLE 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 MATHEMATICA g = 1/Sqrt[1-x]; g[n_] := g[n] = 2x*D[g[n-1], x]; p[n_] := g[n] / g^(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 *) CROSSREFS 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. [From Peter Bala, Apr 03 2011] Cf. A185410 (row reverse), A185411. Sequence in context: A220922 A220993 A219002 * A179077 A038195 A212770 Adjacent sequences:  A156916 A156917 A156918 * A156920 A156921 A156922 KEYWORD easy,nonn,tabl AUTHOR Johannes W. Meijer, Feb 20 2009, Jun 24 2009 EXTENSIONS Minor edits from Johannes W. Meijer, Sep 27 2011 STATUS approved

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Last modified April 11 22:01 EDT 2021. Contains 342888 sequences. (Running on oeis4.)