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A156919
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Table of coefficients of polynomials related to the Dirichlet eta function.
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21
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| The SF(z; n) formulae, see below, were discovered while studying certain properties of the Dirichlet eta function.
From Peter Bala, Apr 3 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)
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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]
Weisstein, Eric W., Dirichlet Eta Function
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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). [From Paul Barry, Jan 26 2011]
From Peter Bala , Apr 3 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]
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EXAMPLE
| 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);
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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
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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.
A186695, A187075. [From Peter Bala , Apr 3 2011]
Sequence in context: A137634 A100229 A071949 * A179077 A038195 A205855
Adjacent sequences: A156916 A156917 A156918 * A156920 A156921 A156922
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009, Jun 24 2009
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EXTENSIONS
| Minor edits from Johannes W. Meijer, Sep 27 2011.
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