A002890 -id:A002890 - cp's OEIS frontend

A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(xt)(1 - 6t + 9t^2 - 4t^3 + t^4)/(4t - 1)/(2*t - 1).

1, 0, 1, 2, 0, 1, 12, 6, 0, 1, 120, 48, 12, 0, 1, 1680, 600, 120, 20, 0, 1, 31680, 10080, 1800, 240, 30, 0, 1, 766080, 221760, 35280, 4200, 420, 42, 0, 1, 22579200, 6128640, 887040, 94080, 8400, 672, 56, 0, 1, 778014720, 203212800, 27578880, 2661120, 211680, 15120, 1008, 72, 0, 1

Offset: 1

Roger L. Bagula, Apr 23 2008

Row sums:
{1, 1, 3, 19, 181, 2421, 43831, 1027783, 29698089, 1011695401, 39319102891}
The t's here are actually Sqrt[] of the variables that give Gamma(1,t) in the Hill reference and is the expansion of Plouffe's rational polynomial for A002890. So this result is related closely to Hill's Gamma(x,y) and seems to be a generalization of the A002890 polynomial.

			Triangle begins:
  {1},
  {0, 1},
  {2, 0, 1},
  {12, 6, 0, 1},
  {120, 48, 12, 0, 1},
  {1680, 600, 120, 20, 0, 1},
  {31680, 10080, 1800, 240, 30, 0, 1},
  {766080, 221760, 35280, 4200, 420, 42, 0, 1},
  {22579200, 6128640, 887040, 94080, 8400, 672, 56, 0, 1},
  {778014720, 203212800, 27578880, 2661120, 211680, 15120, 1008, 72, 0, 1},
  ...

Terrel L. Hill, Statistical Mechanics: Principles and Selected Applications, Dover, New York, 1956, page 336 ff

Cf. A002890, A136264.

Clear[p, f, g] p[t_] = Exp[x*t]*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1); Table[ ExpandAll[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}] a = Table[ CoefficientList[n!*SeriesCoefficient[; FullSimplify[Series[p[t], {t, 0, 30}]], n], x], {n, 0, 10}]; Flatten[a]

p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1) = Sum_{n>=0} P(x,n)*t^n/n!; out_n,m=n!*Coefficients(P(x,n)).

A002891 Low temperature series for spin-1/2 Ising partition function on 3-dimensional simple cubic lattice.

Original entry on oeis.org

1, 0, 0, 1, 0, 3, -3, 15, -30, 101, -261, 807, -2308, 7065, -21171, 65337, -200934, 627249, -1962034, 6192066, -19610346, 62482527, -199807110, 641837193, -2068695927, 6691611633, -21710041944, 70645706963, -230488840446, 753903842400, -2471624380458, 8120879664294, -26736570257010

Offset: 0

N. J. A. Sloane, C. Vohwinkel

sign

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Andrén, Series expansion for the density of states of the Ising and Potts models, arXiv:0706.3116 [cond-mat.str-el], 2007.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
A. J. Guttmann and I. G. Enting, Series studies of the Potts model: I. The simple cubic Ising model, J. Phys. A 26 (1993) 807-821; arXiv:hep-lat/9212032, 1992.
A. J. Wakefield, Statistics of the simple cubic lattice, Proc. Cambridge Philos. Soc. 47 (1951) 419-435 and 799-810.

Cf. A002926 (ferromagnetic susceptibility), A002915 (antiferromagnetic susceptibility), A001393 (high-temperature), A002890 (square lattice), A002892 (f.c.c. lattice), A030045 (4D cubic), A030047 (5D cubic).

Corrections and updates from Steven Finch
"Free energy" changed back to "partition function" (basically the exponential of the free energy) in the name by Andrey Zabolotskiy, Feb 12 2022
a(28)-a(32) added by Andrey Zabolotskiy, Jun 30 2022 using Andrén's data (see his Table 2, column a_n for the coefficients of the expansion of the logarithm of the g.f. of this sequence)

A002927 Low temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

Original entry on oeis.org

0, 0, 1, 8, 60, 416, 2791, 18296, 118016, 752008, 4746341, 29727472, 185016612, 1145415208, 7059265827, 43338407712, 265168691392, 1617656173824, 9842665771649, 59748291677832, 361933688520940, 2188328005246304, 13208464812265559, 79600379336505560, 479025509574159232

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

The zero-field susceptibility per spin is 4m^2/kT * Sum_{n >= 0} a(n) * u^n, where u = exp(-4J/kT). (m is the magnetic moment of a single spin; this factor may be present or absent depending on the precise definition of the susceptibility.) The b-file has been obtained from the series by Guttmann and Jensen via the substitution r = u/(1-u)^2 and dividing by 4. - Andrey Zabolotskiy, Feb 11 2022

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..1305
R. J. Baxter and I. G. Enting, Series expansions for corner transfer matrices: the square lattice Ising model, J. Stat. Physics 21 (1979) 103-123.
C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
I. G. Enting, A, J. Guttmann and I. Jensen, Low-Temperature Series Expansions for the Spin-1 Ising Model, arXiv:hep-lat/9410005, 1994; J. Phys. A. 27 (1994) 6987-7005.
J. W. Essam and M. E. Fisher, Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point, J. Chem. Phys., 38 (1963), 802-812.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
Tony Guttmann, Homepage. See Numerical Data, Ising square lattice susceptibility series, Low temperature series.
Iwan Jensen, Series for the Ising model

Cf. A002906 (high-temperature), A002979 (antiferromagnetic susceptibility), A029872 (specific heat), A002928 (magnetization), A002890 (partition function), A047709 (hexagonal lattice), A002912 (honeycomb), A002926 (cubic lattice), A010115 (spin-1 Ising).

a(n) ~ c * n^(3/4) * (1 + sqrt(2))^(2*n), where c = 0.0187325517235678... - Vaclav Kotesovec, May 06 2024

Corrections and updates from Steven Finch

a(0) = a(1) = 0 prepended, terms a(20) and beyond added by Andrey Zabolotskiy, Feb 10 2022

A029872 Low temperature series for spin-1/2 Ising specific heat on 2D square lattice.

Original entry on oeis.org

16, 72, 288, 1200, 5376, 25480, 125504, 634608, 3269680, 17086168, 90282240, 481347152, 2585485504, 13974825960, 75941188736, 414593263952, 2272626444528, 12502223573304, 68996534259040, 381858968527680, 2118806030647328, 11783826597027256, 65674579024955904

Offset: 0

Steven Finch

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

G. A. Baker, Further application of the Padé approximant method to the Ising and Heisenberg models, Phys. Rev. 129 (1963) 99-102.
I. G. Enting, A, J. Guttmann and I. Jensen, Low-Temperature Series Expansions for the Spin-1 Ising Model, arXiv:hep-lat/9410005, 1994; J. Phys. A. 27 (1994) 6987-7006.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
Index entries for sequences related to specific heat

Cf. A002890 (partition function).

Equals A029873/4 or A029874*8.

CoefficientList[Series[1/(Pi*x^2*(-1 + x^2)^2) * (-2*Pi*x*(1 + x)^2 - (1 + x)^4 * EllipticE[16*(-1 + x)^2*x/(1 + x)^4] + (1 + 30*x^2 + x^4) * EllipticK[16*(-1 + x)^2*x/(1 + x)^4]), {x, 0, 25}], x] (* Vaclav Kotesovec, Apr 28 2024 *)

G.f.: ((u^4 + 30*u^2 + 1) * K(x) / Pi - (u+1)^4 * E(x) / Pi - 2*u*(u+1)^2) / (u^2 * (u^2-1)^2) = 4 * (f(u) * (f'(u)/u + f''(u)) - (f'(u))^2) / f(u)^2, where f(u) is the g.f. of A002890, K(x) and E(x) are the complete elliptic integrals, x = 4*(1-u)*sqrt(u)/(1+u)^2. - Andrey Zabolotskiy, Feb 15 2022

a(n) ~ 2 * (1 + sqrt(2))^(2*n+4) / (Pi*n). - Vaclav Kotesovec, Apr 28 2024

Terms a(18) and beyond from Andrey Zabolotskiy, Feb 15 2022

A370953 Numerators of coefficients of the partition function per spin, lambda (divided by 2), in the very high temperature region, expressed as a power series in the parameter K^2, for the spin-1/2 Ising model on square lattice.

Original entry on oeis.org

1, 1, 4, 77, 1009, 101627, 1302779, 2513121979, 11291682179, 1354947005798, 23064317580681848, 20189102649892270054, 776220757551441546419, 641273428219629914673014, 5433381672262390009892530636, 1399751922597075578762073697769

Offset: 0

N. J. A. Sloane, Mar 10 2024

Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60 (1941), 252-262.
Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part II. Phys. Rev. 60 (1941), 263-276. See (41), p. 263.
Hendrik A. Kramers and Gregory H. Wannier, Extract from page 263 of Part II.
Gandhimohan M. Viswanathan, The hypergeometric series for the partition function of the 2D Ising model, J. Stat. Mech. (2015) P07004; arXiv:1411.2495 [cond-mat.stat-mech], 2014-2015.
Wikipedia, Square lattice Ising model.

See A370954 for denominators.

Cf. A370955, A002908, A002890.

CoefficientList[With[{nmax = 7}, Exp[-Log[2]/2 + 1/(2 Pi) Integrate[Log[Cosh[2k]^2 + Sqrt[Sinh[2k]^4 + 1 - 2 Sinh[2k]^2 Cos[2\[Theta]] + O[k]^(2nmax+1)]], {\[Theta], 0, Pi}] + O[k]^(2nmax+1)]], k][[;; ;; 2]] // Numerator (* Andrey Zabolotskiy, Mar 10 2024 *)
CoefficientList[Cosh[2k] Exp[-x HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16x] /. {x -> (Sinh[2k]/(2Cosh[2k]^2))^2}] + O[k]^32, k][[;; ;; 2]] // Numerator (* Andrey Zabolotskiy, Mar 13 2024, using the g. f. from Gandhimohan M. Viswanathan *)

a(n) / A370954(n) ~ c * 2^(2*n) / (n^3 * log(1 + sqrt(2))^(2*n)), where c = 0.15662885... - Vaclav Kotesovec, May 02 2024

Terms a(5) and beyond from Andrey Zabolotskiy, Mar 10 2024

A370955 Coefficients of the partition function per spin, x(k) (divided by 2), in the low temperature region, expressed as a power series in the parameter k^2, for the spin-1/2 Ising model on square lattice.

Original entry on oeis.org

1, -1, -4, -29, -265, -2745, -30773, -364315, -4488749, -57020414, -741999700, -9845906898, -132774990781, -1814964497342, -25098172218816, -350548840292011, -4938909144117611, -70118741489312657, -1002259422501603334, -14412940220878338617, -208393139882994584383

Offset: 0

N. J. A. Sloane, Mar 10 2024

Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60 (1941), 252-262.
Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part II. Phys. Rev. 60 (1941), 263-276. See (45), p. 264.
Hendrik A. Kramers and Gregory H. Wannier, Extract 1 from page 264 of Part II.
Hendrik A. Kramers and Gregory H. Wannier, Extract 2 from page 264 of Part II.
Gandhimohan M. Viswanathan, The hypergeometric series for the partition function of the 2D Ising model, J. Stat. Mech. (2015) P07004; arXiv:1411.2495 [cond-mat.stat-mech], 2014-2015.
Wikipedia, Square lattice Ising model.

Cf. A002890, A370953, A370954.

CoefficientList[Exp[-x HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16 x]] + O[x]^20, x] (* Andrey Zabolotskiy, Mar 10 2024, using the g. f. from Gandhimohan M. Viswanathan *)

From Vaclav Kotesovec, Apr 28 2024: (Start)
a(n) ~ -c * 16^n / n^2, where c = 0.071286406...
Conjecture: c = exp(2*G/Pi)/(8*Pi) = 0.071286406674269408358123..., where G is the Catalan's constant A006752. (End)

Terms a(6) and beyond from Andrey Zabolotskiy, Mar 10 2024

A260784 Coefficients in a certain low-temperature series expansion.

Original entry on oeis.org

0, 24, 1440, 181440, 43545600, 17882726400, 11333177856000, 10257397742592000, 12540115964952576000, 19887027595237490688000, 39679473692005106319360000, 97249082487667949725286400000, 287164491478121796028858368000000, 1005464789964467723115455053824000000

Offset: 1

N. J. A. Sloane, Aug 04 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..196
Grzegorz Siudem, Agata Fronczak, Bell polynomials in the series expansions of the Ising model, arXiv:2007.16132 [math-ph], 2020.
G. Siudem, A. Fronczak, P. Fronczak, Exact low-temperature series expansion for the partition function of the two-dimensional zero-field s= 1/2 Ising model on the infinite square lattice, arXiv preprint arXiv:1410.7963, 2014. See equations (8) and (11).

Cf. A002890.

A260784 := proc(n)
    local a,d1,d2,d3,d4,d33half ;
    a := 0 ;
    for d2 from 0 do
        if 2*d2 > n then
            break;
        end if;
        for d3 from 0 do
            if 2*d2 +3*d3 > n then
                break;
            end if;
            for d4 from 0 do
                if 2*d2 +3*d3+4*d4 > n then
                    break;
                end if;
                d1 := n-2*d2-3*d3-4*d4 ;
                if d1 >= 0 and type(d1+d3,'even') then
                    d13half := (d1+d3)/2 ;
                    a := a+(d1+d2+d3+d4)!/d1!/d2!/d3!/d4!*(-1)^(d2+d3+d4-1)*2^d2
                        /(d1+d2+d3+d4)*binomial(d1+d3,d13half)^2 ;
                end if;
            end do:
        end do:
    end do:
    a*n!/2 ;
end proc:
seq(A260784(2*n),n=1..15) ; # R. J. Mathar, Aug 27 2015

"Listing 1" in Siudem et al. (2014) gives Mathematica code for the fractions a(n)/(2n)!.

a(n) ~ 2^(2*n) * (1 + sqrt(2))^(2*n) * n^(2*n - 5/2) / (sqrt(Pi) * exp(2*n)). - Vaclav Kotesovec, May 03 2024

cp's OEIS Frontend

A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(x*t)*(1 - 6*t + 9*t^2 - 4*t^3 + t^4)/(4*t - 1)/(2*t - 1).

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A002891 Low temperature series for spin-1/2 Ising partition function on 3-dimensional simple cubic lattice.

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A002927 Low temperature series for spin-1/2 Ising magnetic susceptibility on 2D square lattice.

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A029872 Low temperature series for spin-1/2 Ising specific heat on 2D square lattice.

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A370953 Numerators of coefficients of the partition function per spin, lambda (divided by 2), in the very high temperature region, expressed as a power series in the parameter K^2, for the spin-1/2 Ising model on square lattice.

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Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A370955 Coefficients of the partition function per spin, x(k) (divided by 2), in the low temperature region, expressed as a power series in the parameter k^2, for the spin-1/2 Ising model on square lattice.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A260784 Coefficients in a certain low-temperature series expansion.

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Author

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Maple

Mathematica

Formula

A137514 A triangular sequence from umbral calculus expansion of Simon Plouffe's rational polynomial for A002890: p(x,t) = exp(xt)(1 - 6t + 9t^2 - 4t^3 + t^4)/(4t - 1)/(2*t - 1).