A002895 -id:A002895 - cp's OEIS frontend

A291284 Primes p such that p does not divide any term of the Apery-like sequence A290576.

2, 5, 7, 13, 17, 19, 29, 37, 43, 47, 59, 61, 67, 71, 83, 89, 101, 109, 127, 139, 149, 167, 173, 191, 211, 233, 239, 241, 251, 257, 271, 277, 281, 307, 311, 313, 331, 337, 347, 349, 353, 359, 373, 379, 383, 409, 419, 421, 431, 433, 443

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A002899 Number of n-step polygons on f.c.c. lattice.

Original entry on oeis.org

1, 0, 12, 48, 540, 4320, 42240, 403200, 4038300, 40958400, 423550512, 4434978240, 46982827584, 502437551616, 5417597053440, 58831951546368, 642874989479580, 7063600894137216, 77991775777488144, 864910651813116480

Offset: 0

N. J. A. Sloane

a(n) is the number of 2 X n matrices with entries from {1,2,3,4}, with (1) second row a (multiset) permutation of the first, and (2) no constant columns. - David Callan, Aug 25 2009

a(n) is the constant coefficient in the expansion of (x + y + z + 1/x + 1/y + 1/z + x/y + y/z + z/x + y/x + z/y + x/z)^n. - Seiichi Manyama, Oct 26 2019

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).

Christoph Koutschan, Table of n, a(n) for n = 0..931
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.
Index entries for sequences related to f.c.c. lattice

Cf. A002895, A002898.

f[n_] := Sum[ Binomial[n, k]*(-4)^(n - k)*Sum[ Binomial[k, j]^2*Binomial[2k - 2j, k - j]*Binomial[2j, j], {j, 0, k}], {k, 0, n}]; Array[f, 20, 0]

{a(n)=sum(k=0, n, binomial(n, k)*(-4)^(n-k)*sum(j=0, k, binomial(k, j)^2*binomial(2*k-2*j, k-j)*binomial(2*j, j)))};
print(vector(20, n, a(n-1))) \\ David Broadhurst, Feb 06 2008; fixed by Vaclav Kotesovec, Apr 08 2016

G.f.: hypergeom([1/6, 1/3],[1],108*x^2*(4*x+1))^2. - Mark van Hoeij, Oct 29 2011
Recurrence: n^3*a(n) - 2*n*(2*n-1)*(n-1)*a(n-1) - 16*(n-1)*(5*n^2-10*n+6)*a(n-2) - 96*(n-1)*(n-2)*(2*n-3)*a(n-3) = 0. - R. J. Mathar, Dec 10 2013
a(n) ~ 2^(2*n-2) * 3^(n+3/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 08 2016

More terms from David Broadhurst, Feb 06 2008

A287316 Square array A(n,k) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 93, 70, 1, 0, 1, 6, 45, 256, 639, 252, 1, 0, 1, 7, 66, 545, 2716, 4653, 924, 1, 0, 1, 8, 91, 996, 7885, 31504, 35169, 3432, 1, 0, 1, 9, 120, 1645, 18306, 127905, 387136, 272835, 12870, 1, 0

Offset: 0

Peter Luschny, May 23 2017

A287314 provide polynomials and A287315 rational functions generating the columns of the array.

			Arrays start:
k\n| 0  1    2      3         4        5          6           7
---|----------------------------------------------------------------
k=0| 1, 0,   0,      0,       0,       0,         0,          0, ... A000007
k=1| 1, 1,   1,      1,       1,       1,         1,          1, ... A000012
k=2| 1, 2,   6,     20,      70,     252,       924,       3432, ... A000984
k=3| 1, 3,  15,     93,     639,    4653,     35169,     272835, ... A002893
k=4| 1, 4,  28,    256,    2716,   31504,    387136,    4951552, ... A002895
k=5| 1, 5,  45,    545,    7885,  127905,   2241225,   41467725, ... A169714
k=6| 1, 6,  66,    996,   18306,  384156,   8848236,  218040696, ... A169715
k=7| 1, 7,  91,   1645,   36715,  948157,  27210169,  844691407, ...
k=8| 1, 8, 120,   2528,   66424, 2039808,  70283424, 2643158400, ... A385286
k=9| 1, 9, 153,   3681,  111321, 3965409, 159700401, 7071121017, ...
       A000384,A169711, A169712, A169713,                            A033935

Jeremy Tan, Antidiagonals n = 0..50, flattened
Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.
Ryan S. Bennink, Counting Abelian Squares for a Problem in Quantum Computing, arXiv:2208.02360 [quant-ph], 2022.
Jonathan M. Borwein, A short walk can be beautiful, preprint, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, preprint, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 4-21.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, preprint, FPSAC 2010, San Francisco, USA.
L. Bruce Richmond and Jeffrey Shallit, Counting Abelian Squares, arXiv:0807.5028 [math.CO], 2008.
Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012.

Rows: A000007 (k=0), A000012 (k=1), A000984 (k=2), A002893 (k=3), A002895 (k=4), A169714 (k=5), A169715 (k=6), A385286 (k=8).
Columns: A001477(n=1), A000384 (n=2), A169711 (n=3), A169712 (n=4), A169713 (n=5).
Cf. A033935 (diagonal), A287314, A287315, A287318.

A287316_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((i!)^2*coeff(ser,x,i), i=0..len-1) end:
for k from 0 to 6 do A287316_row(k, 9) od;
A287316_col := proc(n, len) local k, x;
sum(z^k/k!^2, k = 0..infinity); series(%^x, z=0, n+1):
unapply(n!^2*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287316_col(n, 9) od;

Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (n!)^2, {n, 0, 6}], {k, 0,9}]

A287316_row(K, N) = {
  my(x='x + O('x^(2*N-1)));
  Vec(serlaplace(serlaplace(substpol(besseli(0,2*x)^K, 'x^2, 'x))));
};
N=8; concat([vector(N, n, n==1)], vector(9, k, A287316_row(k, N))) \\ Gheorghe Coserea, Jan 12 2018

{A(n, k) = if(n<0 || k<0, 0, n!^2 * polcoeff(besseli(0, 2*x + x*O(x^(2*n)))^k, 2*n))}; /* Michael Somos, Dec 30 2021 */

A(k, n) = my(x='x+O('x^(n+1))); n!^2*polcoeff(hypergeom([], [1], x)^k, n); \\ Peter Luschny, Jun 24 2025

from math import comb
from functools import lru_cache
@lru_cache(maxsize=None)
def A(n,k):
    if k <= 0: return 0**n
    return sum(A(i,k-1)*comb(n,i)**2 for i in range(n+1))
for k in range(10): print([A(n, k) for n in range(8)])
# Jeremy Tan, Dec 10 2021

A(n,k) = A287318(n,k) / binomial(2*n,n).

If a+b=k then A(n,k) = Sum_{i=0..n} A(i,a)*A(n-i,b)*binomial(n,i)^2 (Richmond and Shallit). In particular A(n,k) = Sum_{i=0..n} A(i,k-1)*binomial(n,i)^2. - Jeremy Tan, Dec 10 2021

A039699 Number of 4-dimensional cubic lattice walks that start and end at the origin after 2n steps, free to pass through origin at intermediate stages.

Original entry on oeis.org

1, 8, 168, 5120, 190120, 7939008, 357713664, 16993726464, 839358285480, 42714450658880, 2225741588095168, 118227198981126144, 6380762273973278464, 349019710593278412800, 19310744204362333900800, 1079054103459778710405120, 60818479243449308702049960

Offset: 0

Alessandro Zinani (alzinani(AT)tin.it)

Generating function G(x) is D-finite with a singular point at x = 1/64 (cf. Graph Link). After summing 300000 terms, G(1/64) = 1.239466... and 1 - 1/G(1/64) = 0.193201... Convergence to A086232 is very slow. - Bradley Klee, Aug 20 2018

a(n) is also the constant term in the expansion of (w + 1/w + x + 1/x + y + 1/y + z + 1/z)^(2n). This follows directly from the sequence name, each variable corresponding to a single step in one of the four axis directions. - Christopher J. Smyth, Sep 28 2018

			a(5)=7939008, i.e., there are 7939008 different walks that start and end at origin of a 4-dimensional integer lattice after 2*5=10 steps, free to pass through origin at intermediate steps.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.

Seiichi Manyama, Table of n, a(n) for n = 0..557
S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice. [broken link]
Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice. [Cached copy, with permission of the author]
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 pp. 33, 44.
Bradley Klee, Graph of g.f.
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.
J. Novak, Pólya's random walk theorem, arXiv:1301.3916 [math.PR], 2013.

1-dimensional, 2-dimensional, 3-dimensional analogs are A000984, A002894, A002896. Pólya Constant: A086232.

Row k=4 of A287318.

A039699 := n -> binomial(2*n,n)^2*hypergeom([1/2, -n, -n, -n],[1, 1, 1/2 - n], 1):
seq(simplify(A039699(n)), n=0..14); # Peter Luschny, May 23 2017

max = 30 (* must be even *); Partition[ CoefficientList[ Series[ BesselI[0, 2 x]^4, {x, 0, max}], x]*Range[0, max]!, 2][[All, 1]] (* Jean-François Alcover, Oct 05 2011 *)
With[{nn=30},Take[CoefficientList[Series[BesselI[0,2x]^4,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Aug 09 2013 *)
RecurrenceTable[{256*(n-1)^2*(2*n-3)*(2*n-1)*a[n-2] - 4*(2*n-1)^2*(5*n^2-5*n+2)*a[n-1] + n^4*a[n]==0, a[0]==1, a[1]==8}, a, {n,0,100}] (* Bradley Klee, Aug 20 2018 *)

C=binomial;
A002895(n) = sum(k=0,n, C(n,k)^2 * C(2*n-2*k,n-k) * C(2*k,k) );
a(n)= C(2*n,n) * A002895(n);
/* Joerg Arndt, Apr 19 2013 */

from math import comb
def A039699(n): return comb(n<<1,n)*((sum(comb(n,k)**2*comb(n-k<<1,n-k)*comb(m:=k<<1,k) for k in range(n+1>>1))<<1) + (0 if n&1 else comb(n,n>>1)**4)) # Chai Wah Wu, Jun 17 2025

E.g.f.: Sum_{n>=0} a(2*n) * x^(2*n)/(2*n)! = I_0(2*x)^4. (I = Modified Bessel function of the first kind).
a(n) = binomial(2*n,n)*A002895(n). - Mark van Hoeij, Apr 19 2013
a(n) = binomial(2*n,n)^2*hypergeom([1/2,-n,-n,-n],[1,1,1/2-n],1). - Peter Luschny, May 23 2017
a(n) ~ 2^(6*n+1) / (Pi*n)^2. - Vaclav Kotesovec, Nov 13 2017
From Bradley Klee, Aug 20 2018: (Start)
G.f.: Define G(x) = Sum_{n>=0} a(n)*x^n and G^(j) = (d/dx)^j G(x), then Sum_{j=0..4,k=0..5} M_{j,k}*G^(j)*x^k = 0, with
M={{-8, 768, 0, 0, 0, 0}, {1, -424, 14592, 0, 0, 0}, {0, 7, -1172, 25344, 0, 0}, {0, 0, 6, -640, 10240, 0}, {0, 0, 0, 1, -80, 1024}}.
Sum_{j=0..2,k=0..4} M_{j,k}*a(n-j)*n^k = 0, with
M={{0, 0, 0, 0, 1}, {-8, 52, -132, 160, -80}, {768, -3584, 5888, -4096, 1024}}.
(End)
a(n) = Sum_{i+j+k+l=n, 0<=i,j,k,l<=n} multinomial(2n [i,i,j,j,k,k,l,l]). - Shel Kaphan, Jan 16 2023

A169714 The function W_5(2n) (see Borwein et al. reference for definition).

Original entry on oeis.org

1, 5, 45, 545, 7885, 127905, 2241225, 41467725, 798562125, 15855173825, 322466645545, 6687295253325, 140927922498025, 3010302779775725, 65046639827565525, 1419565970145097545, 31249959913055650125, 693192670456484513025

Offset: 0

N. J. A. Sloane, Apr 17 2010

nonn

Row sums of the fourth power of A008459. - Peter Bala, Mar 05 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. M. Borwein, A short walk can be beautiful, 2015.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals (2012)
Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012

Cf. A002893, A002895, A169715.

A169714 := proc(n)
    W(5,2*n) ;
end proc: # with W() from A169715, R. J. Mathar, Mar 27 2012

a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^5, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 30 2013, after Peter Bala *)
max = 17; Total /@ MatrixPower[Table[Binomial[n, k]^2, {n, 0, max}, {k, 0, max}], 4] (* Jean-François Alcover, Mar 24 2015, after Peter Bala *)

Sum_{n>=0} a(n)*x^n/n!^2 = (Sum_{n>=0} x^n/n!^2)^5 = BesselI(0, 2*sqrt(x))^5. - Peter Bala, Mar 05 2013
D-finite with recurrence: n^4*a(n) = (35*n^4 - 70*n^3 + 63*n^2 - 28*n + 5)*a(n-1) - (n-1)^2*(259*n^2 - 518*n + 285)*a(n-2) + 225*(n-2)^2*(n-1)^2*a(n-3). - Vaclav Kotesovec, Mar 09 2014
a(n) ~ 5^(2*n+5/2) / (16 * Pi^2 * n^2). - Vaclav Kotesovec, Mar 09 2014

A169715 The function W_6(2n) (see Borwein et al. reference for definition).

Original entry on oeis.org

1, 6, 66, 996, 18306, 384156, 8848236, 218040696, 5651108226, 152254667436, 4229523740916, 120430899525096, 3499628148747756, 103446306284890536, 3102500089343886696, 94219208840385966096, 2892652835496484004226, 89662253086458906345036

Offset: 0

N. J. A. Sloane, Apr 17 2010

nonn

Row sums of the fifth power of A008459. - Peter Bala, Mar 05 2013

a(n)/6^(2n) is the probability that two throws of n 6-sided dice will give the same result - Henry Bottomley, Aug 30 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Bernstein and T. Lange, Two grumpy giants and a baby, in ANTS X, Proc. Tenth Algorithmic Number Theory Symposium, 2013.
J. M. Borwein, A short walk can be beautiful, preprint, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, FPSAC 2010, San Francisco, USA.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, preprint, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 4-21.
Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012

Cf. A002893, A002895, A008459, A169714.

W := proc(n,s)
    local a,ai ;
    if s = 0 then
        return 1;
    end if;
    a := 0 ;
    for ai in combinat[partition](s/2) do
        if nops(ai) <= n then
            af := [op(ai),seq(0,i=1+nops(ai)..n)] ;
            a := a+combinat[numbperm](af)*(combinat[multinomial](s/2,op(ai)))^2 ;
        end if ;
    end do;
    a ;
end proc:
A169715 := proc(n)
    W(6,2*n) ;
end proc: # R. J. Mathar, Mar 27 2012

a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^6, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 30 2013, after Peter Bala *)
max = 17; Total /@ MatrixPower[Table[Binomial[n, k]^2, {n, 0, max}, {k, 0, max}], 5] (* Jean-François Alcover, Mar 24 2015, after Peter Bala *)

Sum_{n>=0} a(n)*x^n/n!^2 = (Sum_{n>=0} x^n/n!^2)^6 = BesselI(0, 2*sqrt(x))^6. - Peter Bala, Mar 05 2013
Recurrence: n^5*a(n) = 2*(2*n-1)*(14*n^4 - 28*n^3 + 28*n^2 - 14*n + 3)*a(n-1) - 4*(n-1)^3*(196*n^2 - 392*n + 255)*a(n-2) + 1152*(n-2)^2*(n-1)^2*(2*n-3)*a(n-3). - Vaclav Kotesovec, Mar 09 2014
a(n) ~ 3^(2*n+3) * 4^(n-1) / (Pi*n)^(5/2). - Vaclav Kotesovec, Mar 09 2014

A271673 Number of n-step excursions on the 10-dimensional f.c.c. lattice.

Original entry on oeis.org

1, 0, 180, 5760, 355860, 24226560, 1923670800, 169658496000, 16291413249300, 1674631754611200, 181989927592033680, 20709782925396364800, 2449425950787336166800, 299337868552812779289600, 37621311095831818078152000

Offset: 0

Christoph Koutschan, Apr 12 2016

a(n) = number of walks in the integer lattice Z^10 starting and ending at the origin, using only the steps of the form (s_1, ..., s_10) with s_1^2 + ... + s_10^2 = 2, i.e., each possible step has precisely two nonzero entries which can be +1 or -1.

			There is one walk with no steps.
No walk with a single steps returns to the origin.
The number of returning walks with two steps is exactly the number of allowed steps (called the coordination number of the lattice): a(2) = 4*binomial(10,2).

Christoph Koutschan, Table of n, a(n) for n = 0..449
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green Functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, arXiv:1601.05657 [math-ph], 2016.
S. Hassani, C. Koutschan, J-M. Maillard, N. Zenine, Lattice Green functions: the d-dimensional face-centred cubic lattice, d = 8, 9, 10, 11, 12, Journal of Physics A: Mathematical and Theoretical 49(16) (2016), 164003.
C. Koutschan, Computations for higher-dimensional fcc lattices.
C. Koutschan, Differential operator annihilating the generating function.
C. Koutschan, Recurrence equation.

Cf. A002895, A002899 (d = 3, i.e., excursions on the 3-dimensional f.c.c. lattice), A271432 (d = 4), A271650 (d = 5), A271651 (d = 6), A271670 (d = 7), A271671 (d = 8), A271672 (d = 9), this sequence (d = 10), A271674 (d = 11).

nmax := 50: tt := [seq([seq(add(binomial(2*p,p)*binomial(2*j,2*p-n)*binomial(2*n+2*j-2*p,n+j-p), p = floor((n+1)/2)..floor((n+2*j)/2)), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: for d1 from 3 to 10 do tt := [seq([seq(add(binomial(n,p)*add(binomial(2*j,2*q-p)*binomial(2*j+2*p-2*q,j+p-q)*tt[n-p+1,q+1], q = floor((p+1)/2)..floor((p+2*j)/2)), p = 0..n), j = 0..floor((nmax-n)/2))], n = 0..nmax)]: od: [seq(tt[n+1,1], n = 0..nmax)];

nmax = 50; T = Table[Sum[Binomial[2 p, p]*Binomial[2 j, 2 p - n]*Binomial[2 n + 2 j - 2 p, n + j - p], {p, Floor[(n + 1)/2], Floor[(n + 2 j)/2]}], {n, 0, nmax}, {j, 0, Floor[(nmax - n)/2]}]; Do[T = Table[Sum[Binomial[n, p]*Sum[Binomial[2 j, 2 q - p]*Binomial[2 j + 2 p - 2 q, j + p - q]*T[[n - p + 1, q + 1]], {q, Floor[(p + 1)/2], Floor[(p + 2 j)/2]}], {p, 0, n}], {n, 0, nmax}, {j, 0, If[d1 < 10, Floor[(nmax - n)/2], 0]}], {d1, 3, 10}]; First /@ T

a(n) conjecturally satisfies a linear recurrence equation of order 30 with polynomial coefficients of degree 274 (see link above).

The probability generating function P(z) = Sum_{n>=0} a(n)*(z/180)^n is given by the 10-fold integral (1/Pi)^10 Int_{0..Pi} ... Int_{0..Pi} 1/(1-z*lambda_10) dk_1 ... dk_10, where the structure function is defined as lambda_10 = (1/binomial(10,2)) Sum_{i=1..10} Sum_{j=(i+1)..10} cos(k_i)*cos(k_j). The function P(z) conjecturally satisfies a linear ODE of order 22 with polynomial coefficients of degree 300 (see link above).

A288457 Chebyshev coefficients of density of states of diamond lattice.

Original entry on oeis.org

1, -8, -32, 1024, -12800, 90112, -131072, -2097152, -78774272, 3080716288, -49736056832, 407753457664, -222801428480, -19645180411904, -494299196162048, 22797274090307584, -393216908922454016, 3294704322255781888, 1334801068806111232, -228652837223366918144, -4282607861714030428160, 222230748909257887842304

Offset: 0

Yen-Lee Loh, Jun 16 2017

sign

This is the sequence of integers z^n g_n for n=0,2,4,6,... where g_n are the coefficients in the Chebyshev polynomial expansion of the density of states of the diamond lattice (z=4), g(w) = 1 / (Pi*sqrt(1-w^2)) * Sum_{n>=0} (2-delta_n) g_n T_n(w). Here |w| <= 1 and delta is the Kronecker delta.
The Chebyshev coefficients, g_n, are related to the number of walks on the lattice that return to the origin, W_n, as g_n = Sum_{k=0..n} a_{nk} z^{-k} W_k, where z is the coordination number of the lattice and a_{nk} are the coefficients of Chebyshev polynomials such that T_n(x) = Sum_{k=0..n} a_{nk} x^k.
The author was unable to obtain a closed form for z^n g_n.

Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.

Related to numbers of walks returning to origin, W_n, on diamond lattice (A002895).

Wdia[n_] := If[OddQ[n], 0,
   Sum[Binomial[n/2,j]^2 Binomial[2j,j] Binomial[n-2j, n/2-j], {j, 0, n/2}]];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*4^(n-k)*Wdia[k], {k, 0, n}];
Table[zng[n], {n,0,50}]

A362676 a(n) = Sum_{k = 0..n} 4^(n-k)binomial(n,k)binomial(n-1,k)binomial(2k,k).

Original entry on oeis.org

1, 4, 32, 328, 3840, 48504, 641984, 8765712, 122370048, 1736921560, 24975268032, 362872728816, 5317470233088, 78479369810352, 1165299414952320, 17393306836535328, 260791399517110272, 3925811865435871896, 59305018671515758784

Offset: 0

Peter Bala, Jul 03 2023

The sequence of Franel numbers A000172 satisfies the identity A000172(n) = Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*binomial(n+2*k,2*k)*binomial(2*k,k). The present sequence comes from the following modification of the right-hand side of the identity: a(n) = Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*binomial(-n+k,k)* binomial(2*k,k), multipled by a factor (-1)^n to give positive terms.
The Franel numbers satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r. We conjecture that the present sequence satisfies the same supercongruences.
From Peter Bala, Jul 07 2023 (Start):
Compare with the Domb numbers A002895, which are defined by A002895(n) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(2*n-2*k,n-k) * binomial(2*k,k).
The supercongruences A002895(n*p^r) == A002895(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r (see Osburn and Sahu).
We conjecture that the present sequence satisfies the same supercongruences. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..833
Robert Osburn and Brundaban Sahu, A supercongruence for generalized Domb numbers, arXiv:1201.6195v2 [math.NT], Functiones et Approximatio. Comment. Math, Vol. 48, No 1, March 2013, 29-36.

Cf. A000172, A363985 - A363990, A002895, A081085, A364111.

seq(add(4^(n-k)*binomial(n,k)*binomial(n-1,k)*binomial(2*k,k), k = 0..n), n = 0..20);
# alternative faster program for large n
seq(simplify(4^n*hypergeom([-n, 1 - n, 1/2], [1, 1], 1)), n = 0..20);
# alternative (Peter Bala Jul 07 2023)
seq(add(binomial(n+k-1,k) * binomial(2*n-2*k,n-k) * binomial(2*k,k), k = 0..n), n = 0..20);

Table[4^n * HypergeometricPFQ[{-n, 1 - n, 1/2}, {1, 1}, 1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 04 2023 *)

from sympy import hyper, hyperexpand, S
def A362676(n): return int(hyperexpand(hyper((-n, 1-n, S.Half), [1,1], 1))*(1<<(n<<1))) # Chai Wah Wu, Jul 10 2023

a(n) = 4^n * hypergeom ([-n, 1 - n, 1/2], [1, 1], 1).
From Vaclav Kotesovec, Jul 04 2023: (Start)
Recurrence: (n-1)*n^2*(3*n^2 - 9*n + 7)*a(n) = 4*(n-1)*(15*n^4 - 60*n^3 + 80*n^2 - 40*n + 8)*a(n-1) - 4*(n-2)*(4*n - 7)*(4*n - 5)*(3*n^2 - 3*n + 1)*a(n-2).
a(n) ~ 2^(4*n - 1/2) / (Pi*n). (End)
a(n) = Sum_{k = 0..n} (-1)^k * binomial(-n,k) * binomial(2*n-2*k,n-k) * binomial(2*k,k). Cf. A081085. Peter Bala, Jul 07 2023
a(n) = binomial(2*n,n)*hypergeom([-n, n, 1/2], [1, 1/2 - n], 1). - Peter Bala, Jul 07 2023

A169711 The function W_n(6) (see Borwein et al. reference for definition).

Original entry on oeis.org

1, 20, 93, 256, 545, 996, 1645, 2528, 3681, 5140, 6941, 9120, 11713, 14756, 18285, 22336, 26945, 32148, 37981, 44480, 51681, 59620, 68333, 77856, 88225, 99476, 111645, 124768, 138881, 154020, 170221, 187520, 205953, 225556, 246365, 268416, 291745, 316388

Offset: 1

N. J. A. Sloane, Apr 17 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jonathan M. Borwein, Dirk Nuyens, Armin Straub, and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, May 2011.
Pakawut Jiradilok and Elchanan Mossel, Gaussian Broadcast on Grids, arXiv:2402.11990 [cs.IT], 2024. See p. 27.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

The sequence in Table 1 of the Borwein et al. reference are A000384, A109711-A109713; A000984, A002893, A002895, A169714, A169715.
Column 3 of A287316.
Cf. A287314.

[6*n^3-9*n^2+4*n: n in [1..40]]; // Vincenzo Librandi, May 28 2017

A169711 := proc(n)
        W(n,6) ;
end proc:
seq(A169711(n),n=1..20) ; # uses W from A169715; R. J. Mathar, Mar 28 2012
a := n -> 6*n^3 - 9*n^2 + 4*n: seq(a(n), n=1..33); # Peter Luschny, May 27 2017

CoefficientList[Series[(1 + 16 x + 19 x^2) / (1 - x)^4, {x, 0, 50}], x] (* or *) Table[6 n^3 - 9 n^2 + 4 n, {n, 1, 40}] (* Vincenzo Librandi, May 28 2017 *)
LinearRecurrence[{4,-6,4,-1},{1,20,93,256},40] (* Harvey P. Dale, Feb 27 2023 *)

a(n)=6*n^3-9*n^2+4*n \\ Charles R Greathouse IV, Oct 18 2022

a(n) = 6*n^3 - 9*n^2 + 4*n. - Peter Luschny, May 27 2017
G.f.: x*(1+16*x+19*x^2)/(1-x)^4. - Vincenzo Librandi, May 28 2017
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, May 28 2017

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A271673 Number of n-step excursions on the 10-dimensional f.c.c. lattice.

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A362676 a(n) = Sum_{k = 0..n} 4^(n-k)binomial(n,k)binomial(n-1,k)binomial(2k,k).