A138540 -id:A138540 - cp's OEIS frontend

A002896 Number of 2n-step polygons on cubic lattice.

1, 6, 90, 1860, 44730, 1172556, 32496156, 936369720, 27770358330, 842090474940, 25989269017140, 813689707488840, 25780447171287900, 825043888527957000, 26630804377937061000, 865978374333905289360, 28342398385058078078010, 932905175625150142902300

Offset: 0

N. J. A. Sloane

Number of walks with 2n steps on the cubic lattice Z^3 beginning and ending at (0,0,0).
If A is a random matrix in USp(6) (6 X 6 complex matrices that are unitary and symplectic) then a(n) is the 2n-th moment of tr(A^k) for all k >= 7. - Andrew V. Sutherland, Mar 24 2008
Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*x*z + w*y + x*z + y + z)). - Gheorghe Coserea, Jul 14 2016
Constant term in the expansion of (x + 1/x + y + 1/y + z + 1/z)^(2n). - Harry Richman, Apr 29 2020

			1 + 6*x + 90*x^2 + 1860*x^3 + 44730*x^4 + 1172556*x^5 + 32496156*x^6 + ...

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).

Alois P. Heinz, Table of n, a(n) for n = 0..645
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.
Nikolai Beluhov, Powers of 2 in Balanced Grid Colourings, arXiv:2504.21451 [math.CO], 2025.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024. See p. 5.
Li Gan, On the algebraic area of cubic lattice walks, arXiv:2307.08732 [math-ph], 2023.
S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 23.
W. K. Hayman, A Generalisation of Stirling's Formula, Journal für die reine und angewandte Mathematik, (1956), vol. 196, pp. 67-95.
John A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
Heng Huat Chan, Yoshio Tanigawa, Yifan Yang, and Wadim Zudilin, New analogues of Clausen's identities arising from the theory of modular forms, Advances in Mathematics, 228:2 (2011), pp. 1294-1314; doi:10.1016/j.aim.2011.06.011.
G. S. Joyce, The simple cubic lattice Green function, Phil. Trans. Roy. Soc., 273 (1972), 583-610.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.
Nobu C. Shirai and Naoyuki Sakumichi, Universal Negative Energetic Elasticity in Polymer Chains: Crossovers among Random, Self-Avoiding, and Neighbor-Avoiding Walks, arXiv:2408.14992 [cond-mat.soft], 2024. See p. 5.
Chunwei Song and Bowen Yao, On Combinatorial Rectangles with Minimum oo-Discrepancy, arXiv:1909.05648 [math.CO], 2019. See p. 7 for another interpretation.
Ralph A. Raimi and Jet Wimp, Review of book "A=B" by M. Petkovsek et al., Mathematical Intelligencer, 23 (No. 4, 2001), pp. 72-77.

C(2n, n) times A002893.
Cf. A049020, A049037, A084261, A138540, A174516.
Related to diagonal of rational functions: A268545-A268555.
Row k=3 of A287318.

a := proc(n) local k; binomial(2*n,n)*add(binomial(n,k)^2 *binomial(2*k,k), k=0..n); end;
# second Maple program
a:= proc(n) option remember; `if`(n<2, 5*n+1,
      (2*(2*n-1)*(10*n^2-10*n+3) *a(n-1)
       -36*(n-1)*(2*n-1)*(2*n-3) *a(n-2)) /n^3)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 02 2012
A002896 := n -> binomial(2*n,n)*hypergeom([1/2, -n, -n], [1, 1], 4):
seq(simplify(A002896(n)), n=0..16); # Peter Luschny, May 23 2017

Table[Binomial[2n,n] Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 24 2012 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {-n, -n, 1/2}, {1, 1}, 4] Binomial[ 2 n, n]] (* Michael Somos, May 21 2013 *)

a(n)=binomial(2*n,n)*sum(k=0,n,binomial(n, k)^2*binomial(2*k, k)) \\ Charles R Greathouse IV, Oct 31 2011

def A002896():
    x, y, n = 1, 6, 1
    while True:
        yield x
        n += 1
        x, y = y, ((4*n-2)*((10*(n-1)*n+3)*y-18*(n-1)*(2*n-3)*x))//n^3
a = A002896()
[next(a) for i in range(17)]  # Peter Luschny, Oct 09 2013

a(n) = C(2*n, n)*Sum_{k=0..n} C(n, k)^2*C(2*k, k).
a(n) = (4^n*p(1/2, n)/n!)*hypergeom([-n, -n, 1/2], [1, 1], 4), where p(a, k) = Product_{i=0..k-1} (a+i).
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2*x)^3. - Corrected by Christopher J. Smyth, Oct 29 2012
D-finite with recurrence: n^3*a(n) = 2*(2*n-1)*(10*n^2-10*n+3)*a(n-1) - 36*(n-1)*(2*n-1)*(2*n-3)*a(n-2). - Vladeta Jovovic, Jul 16 2004
An asymptotic formula follows immediately from an observation of Bruce Richmond and me in SIAM Review - 31 (1989, 122-125). We use Hayman's method to find the asymptotic behavior of the sum of squares of the multinomial coefficients multi(n, k_1, k_2, ..., k_m) with m fixed. From this one gets a_n ~ (3/4)*sqrt(3)*6^(2*n)/(Pi*n)^(3/2). - Cecil C Rousseau (ccrousse(AT)memphis.edu), Mar 14 2006
G.f.: (1/sqrt(1+12*z)) * hypergeom([1/8,3/8],[1],64/81*z*(1+sqrt(1-36*z))^2*(2+sqrt(1-36*z))^4/(1+12*z)^4) * hypergeom([1/8, 3/8],[1],64/81*z*(1-sqrt(1-36*z))^2*(2-sqrt(1-36*z))^4/(1+12*z)^4). - Sergey Perepechko, Jan 26 2011
a(n) = binomial(2*n,n)*A002893(n). - Mark van Hoeij, Oct 29 2011
G.f.: (1/2)*(10-72*x-6*(144*x^2-40*x+1)^(1/2))^(1/2)*hypergeom([1/6, 1/3],[1],54*x*(108*x^2-27*x+1+(9*x-1)*(144*x^2-40*x+1)^(1/2)))^2. - Mark van Hoeij, Nov 12 2011
PSUM transform is A174516. - Michael Somos, May 21 2013
0 = (-x^2+40*x^3-144*x^4)*y''' + (-3*x+180*x^2-864*x^3)*y'' + (-1+132*x-972*x^2)*y' + (6-108*x)*y, where y is the g.f. - Gheorghe Coserea, Jul 14 2016
a(n) = [(x y z)^0] (x + 1/x + y + 1/y + z + 1/z)^(2*n). - Christopher J. Smyth, Sep 25 2018
a(n) = (1/Pi)^3*Integral_{0 <= x, y, z <= Pi} (2*cos(x) + 2*cos(y) + 2*cos(z))^(2*n) dx dy dz. - Peter Bala, Feb 10 2022
a(n) = Sum_{i+j+k=n, 0<=i,j,k<=n} multinomial(2n [i,i,j,j,k,k]). - Shel Kaphan, Jan 16 2023
Sum_{k>=0} a(k)/36^k = A086231 = (sqrt(3)-1) * (Gamma(1/24) * Gamma(11/24))^2 / (32*Pi^3). - Vaclav Kotesovec, Apr 23 2023
G.f.: HeunG(1/9,1/12,1/4,3/4,1,1/2,4*x)^2 (see Hassani et al.). - Stefano Spezia, Feb 16 2025

A138547 Moment sequence of tr(A^6) in USp(6).

Original entry on oeis.org

1, -1, 6, -15, 90, -310, 1860, -7455, 44730, -195426, 1172556, -5416026, 32496156, -156061620, 936369720, -4628393055, 27770358330, -140348412490, 842090474940, -4331544836190, 25989269017140, -135614951248140, 813689707488840, -4296741195214650, 25780447171287900

Offset: 0

Andrew V. Sutherland, Mar 24 2008

sign

If A is a random matrix in the compact group USp(6) (6 X 6 complex matrices that are unitary and symplectic), then a(n) = E[(tr(A^6))^n] is the n-th moment of the trace of A^6. See A138547 for central moments.

			a(3) = -15 because E[(tr(A^6))^3] = -15 for a random matrix A in USp(6).

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A138540, A138547.

mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(B_{(2j-m)/6}(z)-B_{(2j-m+2)/6}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z) with I_v(z) is the hyperbolic Bessel function (of the first kind) of order v.

A138544 Moment sequence of tr(A^4) in USp(6).

Original entry on oeis.org

1, -1, 4, -9, 42, -130, 660, -2415, 12810, -51786, 281736, -1216446, 6727644, -30440124, 170316432, -798126615, 4504487130, -21692469370, 123255492360, -606672653730, 3465702008340, -17366224451940, 99645553785960, -506814533253210, 2918768920720380, -15034038412333500

Offset: 0

Andrew V. Sutherland, Mar 24 2008

sign

If A is a random matrix in the compact group USp(6) (6 X 6 complex matrices which are unitary and symplectic), then a(n) = E[(tr(A^4))^n] is the n-th moment of the trace of A^4. See A138545 for central moments.

			a(3) = -9 because E[(tr(A^4))^3] = -9 for a random matrix A in USp(6).

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A138540, A138545.

mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(B_{(2j-m)/4}(z)-B_{(2j-m+2)/4}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z), with I_v(z) the hyperbolic Bessel function (of the first kind) of order v.

A138546 Moment sequence of tr(A^5) in USp(6).

Original entry on oeis.org

1, 0, 4, 0, 42, 0, 660, 0, 12810, 0, 281736, 0, 6727644, 0, 170316432, 0, 4504487130, 0, 123255492360, 0, 3465702008340, 0, 99645553785960, 0, 2918768920720380, 0, 86852063374902000, 0, 2619552500788984200, 0, 79939673971478231760, 0

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

If A is a random matrix in the compact group USp(6) (6 X 6 complex matrices that are unitary and symplectic), then a(n) = E[(tr(A^5))^n] is the n-th moment of the trace of A^5.

			a(4) = 42 because E[(tr(A^5))^4] = 42 for a random matrix A in USp(6).

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A138540.

mgf is A(z) = det[F_{i+j-2}(z)], 1<=i,j<=3, where F_m(z) = Sum_j binomial(m,j)(B_{(2j-m)/5}(z)-B_{(2j-m+2)/5}(z)) and B_v(z)=0 for non-integer v and otherwise B_v(z)=I_v(2z) with I_v(z) the hyperbolic Bessel function (of the first kind) of order v.

A138549 Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

Original entry on oeis.org

1, 1, 2, 5, 16, 62, 282, 1459, 8375, 52323, 350676, 2493846, 18659787, 145918295, 1186129168, 9978055080, 86545684565, 771571356565, 7051538798490, 65913863945775, 628919704903746, 6114899366942556, 60492393411513722

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tI-A) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[X^n].
Let L_p(T) be the L-polynomial (numerator of the zeta function) of a genus 3 curve C. Under a generalized Sato-Tate conjecture, for almost all C, a(n) is the n-th moment of the coefficient of t^2 in L_p(t/sqrt(p)), as p varies.
See A138550 for central moments.

			a(3) = 5 because E[X^3] = 5 for X the t^2 coeff of det(tI-A) in USp(6).

Kiran S. Kedlaya, Andrew V. Sutherland, Computing L-series of hyperelliptic curves, arXiv:0801.2778 [math.NT], 2008-2012; Algorithmic Number Theory Symposium--ANTS VIII, 2008.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Nicholas M. Katz and Peter Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS, 1999.

Cf. A138540, A138550, A138356.

See Prop. 12 of first Kedlaya-Sutherland reference.

A138550 Central moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

Original entry on oeis.org

1, 0, 1, 1, 5, 16, 75, 366, 2016, 11936, 75678, 507575, 3575693, 26289408, 200709665, 1584482382, 12888498820, 107698656192, 922140333952, 8072379904752, 72108967554160, 656190909218560, 6074106708205200, 57118680813847840

Offset: 0

Andrew V. Sutherland, Mar 24 2008

nonn

Let the random variable X be the coefficient of t^2 in the characteristic polynomial det(tI-A) of a random matrix in USp(6) (6x6 complex matrices that are unitary and symplectic). Then a(n) = E[(X-1)^n] is the n-th central moment of X since E[X]=1 (see A138549).

Dimension of space of invariant tensors in second fundamental representation of Sp(6). - Bruce Westbury, Dec 05 2014

			a(4) = 5 because E[(X-1)^4] = 5 for X the t^2 coeff of det(tI-A) in USp(6).

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A138540, A138549, A251591.

LiE
```
p_tensor(n,[0,1,0],C3)|[0,0,0]
```

a(n) = Sum_{k=0..n} binomial(n,k)(-1)^{n-k}*A138549(k).

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A136092 Bisection of A138540.

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A002896 Number of 2n-step polygons on cubic lattice.

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A138547 Moment sequence of tr(A^6) in USp(6).

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A138544 Moment sequence of tr(A^4) in USp(6).

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A138546 Moment sequence of tr(A^5) in USp(6).

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A138549 Moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

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A138550 Central moment sequence of t^2 coefficient in det(tI-A) for random matrix A in USp(6).

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A138541 Moment sequence of tr(A^2) in USp(6).

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A138542 Central moment sequence of tr(A^2) in USp(6).

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