A055857 -id:A055857 - cp's OEIS frontend

A055856 Susceptibility series H_4 for 2-dimensional Ising model (divided by 2).

1, 16, 90, 328, 888, 2016, 3994, 7212, 12070, 19112, 28846, 41976, 59116, 81132, 108738, 142972, 184638, 234952, 294806, 365596, 448296, 544492, 655230, 782292, 926794, 1090716, 1275238, 1482548, 1713880, 1971636, 2257102, 2572896, 2920350, 3302308, 3720138

Offset: 0

Wolfdieter Lang, Jun 07 2000

Colin Barker, Table of n, a(n) for n = 0..1000
A. J. Guttmann and I. G. Enting, Solvability of some statistical mechanical systems, Phys. Rev. Lett., 76 (1996), 344-347.
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-4,0,4,2,-3,-1,1).

Cf. A054275, A054410, A054389, A054764, A055857.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1 +15*x +71*x^2 +192*x^3 +326*x^4 +388*x^5 +326*x^6 +192*x^7 + 71*x^8 +15*x^9 +x^10)/((1-x^3)*(1-x^2)^3*(1-x)) )); // G. C. Greubel, Jan 16 2020

1,seq( simplify( (4794*n^4 +19194*n^2 +3349 -81*(-1)^n*(2*n^2 +5) + 512*ChebyshevT(n, -1/2))/1728 ), n=1..40); # G. C. Greubel, Jan 16 2020

Join[{1}, Table[(4794*n^4 +19194*n^2 +3349 -81*(-1)^n*(2*n^2 +5) + 512*ChebyshevT[n, -1/2])/1728, {n,40}]] (* G. C. Greubel, Jan 16 2020 *)
LinearRecurrence[{1,3,-2,-4,0,4,2,-3,-1,1},{1,16,90,328,888,2016,3994,7212,12070,19112,28846},40] (* Harvey P. Dale, Jul 24 2021 *)

Vec((1 +15*x +71*x^2 +192*x^3 +326*x^4 +388*x^5 +326*x^6 +192*x^7 + 71*x^8 +15*x^9 +x^10)/((1-x^3)*(1-x)^4*(1+x)^3) + O(x^40)) \\ Colin Barker, Dec 10 2016

[1]+[(4794*n^4 +19194*n^2 +3349 -81*(-1)^n*(2*n^2 +5) + 512*chebyshev_T(n, -1/2))/1728 for n in (1..40)] # G. C. Greubel, Jan 16 2020

G.f.: (1 + 15*x + 71*x^2 + 192*x^3 + 326*x^4 + 388*x^5 + 326*x^6 + 192*x^7 + 71*x^8 + 15*x^9 + x^10)/((1-x^3)*(1-x)^4*(1+x)^3).

a(n) = (4794*n^4 + 19194*n^2 + 3349 - 81*(-1)^n*(2*n^2 + 5) + 512*ChebyshevT(n, -1/2))/1728, for n >= 1, with a(0) = 1. - G. C. Greubel, Jan 16 2020

A055855 Convolution of A055854 with A011782.

Original entry on oeis.org

0, 1, 10, 64, 328, 1462, 5908, 22180, 78592, 265729, 864146, 2719028, 8316200, 24814832, 72453344, 207502016, 584094080, 1618757120, 4423347200, 11932579840, 31812874240, 83901227008, 219074805760, 566754967552

Offset: 0

Wolfdieter Lang May 30 2000

Tenth column of triangle A055587.

T(n,8) of array T as in A049600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-144,672,-2016,4032,-5376,4608,-2304,512).

Cf. A011782, A049600, A055584, A055857.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-x)^8/(1-2*x)^9 )); // G. C. Greubel, Jan 16 2020

seq(coeff(series(x*(1-x)^8/(1-2*x)^9, x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 16 2020

CoefficientList[Series[x*(1-x)^8/(1-2*x)^9, {x,0,30}], x] (* G. C. Greubel, Jan 16 2020 *)

my(x='x+O('x^30)); concat([0], Vec(x*(1-x)^8/(1-2*x)^9)) \\ G. C. Greubel, Jan 16 2020

def A055855_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)^8/(1-2*x)^9 ).list()
A055855_list(30) # G. C. Greubel, Jan 16 2020

a(n) = T(n, 8) = A055587(n+8, 9).

G.f.: x*(1-x)^8/(1-2*x)^9.

cp's OEIS Frontend

A055856 Susceptibility series H_4 for 2-dimensional Ising model (divided by 2).

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