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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
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OFFSET
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0,2
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COMMENTS
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The SF(z; n) formulas, see below, were discovered while studying certain properties of the Dirichlet eta function.
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
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FORMULA
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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
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.
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EXAMPLE
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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;
...
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MAPLE
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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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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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