A039699 -id:A039699 - cp's OEIS frontend

A385087 2-adic valuation of A039699.

3, 3, 10, 3, 6, 8, 12, 3, 6, 6, 11, 8, 11, 12, 16, 3, 6, 6, 12, 6, 9, 13, 17, 8, 11, 11, 16, 12, 15, 16, 20, 3, 6, 6, 14, 6, 9, 11, 15, 6, 9, 9, 14, 13, 16, 17, 21, 8, 11, 11, 17, 11, 14, 17, 21, 12, 15, 15, 20, 16, 19, 20, 24, 3, 6, 6, 13, 6, 9, 11, 15, 6, 9, 9, 14, 11, 14, 15, 19, 6

Offset: 1

Michel Marcus, Jun 17 2025

Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See w4(n) in Table 1 p. 2.

Cf. A000120, A002895, A007814, A039699.

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) = hammingweight(n) + valuation(A002895(n), 2);

from math import comb
def A385087(n): return (~(a:=(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)) & a-1).bit_length() + n.bit_count() # Chai Wah Wu, Jun 17 2025

a(n) = A007814(A039699(n)) = A000120(n) + A007814(A002895(n)).

A287318 Square array A(n,k) = (2n)! [x^n] BesselI(0, 2sqrt(x))^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 36, 20, 0, 1, 8, 90, 400, 70, 0, 1, 10, 168, 1860, 4900, 252, 0, 1, 12, 270, 5120, 44730, 63504, 924, 0, 1, 14, 396, 10900, 190120, 1172556, 853776, 3432, 0, 1, 16, 546, 19920, 551950, 7939008, 32496156, 11778624, 12870, 0

Offset: 0

Peter Luschny, May 23 2017

			Arrays start:
  k\n| 0   1    2      3        4          5           6
  ---|---------------------------------------------------------
  k=0| 1,  0,   0,     0,       0,         0,            0, ... A000007
  k=1| 1,  2,   6,    20,      70,       252,          924, ... A000984
  k=2| 1,  4,  36,   400,    4900,     63504,       853776, ... A002894
  k=3| 1,  6,  90,  1860,   44730,   1172556,     32496156, ... A002896
  k=4| 1,  8, 168,  5120,  190120,   7939008,    357713664, ... A039699
  k=5| 1, 10, 270, 10900,  551950,  32232060,   2070891900, ... A287317
  k=6| 1, 12, 396, 19920, 1281420,  96807312,   8175770064, ... A356258
  k=7| 1, 14, 546, 32900, 2570050, 238935564,  25142196156, ...
  k=8| 1, 16, 720, 50560, 4649680, 514031616,  64941883776, ...
  k=9| 1, 18, 918, 73620, 7792470, 999283068, 147563170524, ...

Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.

Rows: A000007 (k=0), A000984 (k=1), A002894 (k=2), A002896 (k=3), A039699 (k=4), A287317 (k=5), A356258 (k=6).
Columns: A005843 (n=1), A152746 (n=2), 20*A169711 (n=3), 70*A169712 (n=4), 252*A169713 (n=5).
Main diagonal gives A303503.
Cf. A287316.

A287318_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((2*i)!*coeff(ser,x,i), i=0..len-1) end:
for k from 0 to 6 do A287318_row(k, 9) od;

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

A(n,k) = A287316(n,k) * binomial(2*n,n).

A288459 Chebyshev coefficients of density of states of 4D hypercubic lattice.

Original entry on oeis.org

1, -48, 1344, -24576, 218112, -688128, 926416896, -95932121088, 5186228846592, -154060166529024, 1455620351852544, -29436202608230400, 17834604768232734720, -1968810407797802926080, 114581075578951670169600, -3629224301781687956668416, 33517817437575659447648256, -1040884075746436707891806208

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 four-dimensional hypercubic lattice (z=8), 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 hypercubic lattice (A039699).

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}]];
Whcub[n_] := Binomial[n, n/2] Wdia[n];
ank[n_, k_] := SeriesCoefficient[ChebyshevT[n, x], {x, 0, k}];
zng[n_] := Sum[ank[n, k]*8^(n-k)*Whcub[k], {k, 0, n}];
Table[zng[n], {n,0,50}]

A287317 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

Original entry on oeis.org

1, 10, 270, 10900, 551950, 32232060, 2070891900, 142317232200, 10277494548750, 770878551371500, 59577647564312020, 4717432065143561400, 381091087190569291900, 31308955091335405435000, 2609450031306515140215000, 220199552765301571338488400

Offset: 0

Peter Luschny, May 23 2017

Case k=5 of A287318.
Cf. A169714, A287316
1-4 dimensional analogs are A000984, A002894, A002896, A039699.

A287317_list := proc(len) series(BesselI(0, 2*sqrt(x))^5, x, len);
seq((2*i)!*coeff(%, x, i), i=0..len-1) end: A287317_list(16);

Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^5, {x, 0, n}] (2 n) !, {n, 0, 15}]
Table[Binomial[2n,n]^2 Sum[(Binomial[n,j]^4/Binomial[2n,2j]) HypergeometricPFQ[{-j,-j,-j}, {1,1/2-j}, 1/4], {j,0,n}], {n,0,15}]
Table[Sum[(2 n)!/(i! j! k! l! (n-i-j-k-l)!)^2, {i,0,n}, {j,0,n-i}, {k,0,n-i-j}, {l,0,n-i-j-k}], {n,0,30}] (* Shel Kaphan, Jan 24 2023 *)

a(n) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^5.
a(n) = binomial(2*n,n)*A169714(n).
a(n) ~ 2^(2*n) * 5^(2*n + 5/2) / (16 * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Nov 13 2017
a(n) = Sum_{i+j+k+l+m=n, 0<=i,j,k,l,m<=n} multinomial(2n, [i,i,j,j,k,k,l,l,m,m]). - Shel Kaphan, Jan 24 2023

Moved original definition to formula section and reworded definition descriptively similar to sequence A039699, by Dave R.M. Langers, Oct 12 2022

A303503 a(n) = (2n)! [x^(2n)] BesselI(0,2x)^n.

Original entry on oeis.org

1, 2, 36, 1860, 190120, 32232060, 8175770064, 2898980908824, 1369263687414480, 830988068906518380, 630109741730668410640, 583773362067938664133512, 648851848280206013365243776, 852146184628067383511375555000, 1305460597778526044143501996708800, 2307324514460203126471248458864413200

Offset: 0

Ilya Gutkovskiy, Apr 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..212

Main diagonal of A287318.

Cf. A000984, A002894, A002896, A033935, A039699, A287317, A361297.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> (2*n)!*b(n$2)/n!^2:
seq(a(n), n=0..17);  # Alois P. Heinz, Jan 29 2023

Table[(2 n)! SeriesCoefficient[BesselI[0, 2 x]^n, {x, 0, 2 n}], {n, 0, 15}]

a(n) = A287318(n,n).
a(n) ~ c * d^n * n^(2*n), where c = 1.72802011936236389522137050964080... and d = 1.1381284656425793765251319541847869000364101065484286935... - Vaclav Kotesovec, Apr 26 2018
a(n) = A000984(n)*A033935(n). - Alois P. Heinz, Jan 30 2023

A359801 Number of 4-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

Original entry on oeis.org

1, 8, 104, 2944, 108136, 4525888, 204981888, 9792786432, 486323201640, 24874892400064, 1302278744460352, 69474942954714112, 3764568243058030208, 206675027529594291200, 11473858525271117889536, 643154944963894079717376, 36355546411928157876528744, 2070313613815122857027563200

Offset: 0

Shel Kaphan, Mar 08 2023

In Novak's note it is mentioned that if P(z) and Q(z) are the g.f.s for the probabilities of indecomposable and decomposable loops, respectively, that P(z) = 1 - 1/Q(z). This works equally well using loop counts rather than probabilities. The g.f.s may be expressed by the series constructed from the sequences of counts of loops of length 2*n. Q(z) for the 4-d case is the series corresponding to A039699.

To obtain the probability of returning to the point of origin for the first time after 2*n steps, divide a(n) by the total number of walks of length 2*n in d dimensions: (2*d)^(2*n) = 64^n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..550
Dorin Dumitraşcu and Liviu Suciu, Asymptotics for the Number of Random Walks in the Euclidean Lattice, arXiv:2212.01702 [math.CO], 2022, p.11.
Jonathan Novak, Pólya's Random Walk Theorem, The American Mathematical Monthly, Vol. 121, No. 8 (October 2014), pp. 711-716.

Cf. A039699, A287317 (number of walks that return to the origin in 2n steps).
Number of walks that return to the origin for the first time in 2n steps, in 1..3 dimensions: |A002420|, A054474, A049037.
Column k=4 of A361397.

walk4d[n_] :=
 Sum[(2 n)!/(i! j! k! (n - i - j - k)!)^2, {i, 0, n}, {j, 0,
   n - i}, {k, 0, n - i - j}]; invertSeq[seq_] :=
  CoefficientList[1 - 1/SeriesData[x, 0, seq, 0, Length[seq], 1], x]; invertSeq[Table[walk4d[n], {n, 0, 17}]]

seq(n) = {my(v=Vec(2 - 1/serlaplace(besseli(0, 2*x + O(x^(2*n+1)))^4))); vector(n+1, i, v[2*i-1])} \\ Andrew Howroyd, Mar 08 2023

G.f.: 2 - 1/Q(x) where Q(x) is the g.f. of A039699.
G.f.: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^4 dt.
INVERTi transform of A039699.

A328714 Constant term in the expansion of (1 + w + x + y + z + 1/w + 1/x + 1/y + 1/z)^n.

Original entry on oeis.org

1, 1, 9, 25, 217, 921, 7761, 41889, 345465, 2162617, 17605249, 121120209, 980612161, 7174425025, 58079091513, 442755733065, 3595708057785, 28197440412345, 230133477721665, 1841288167473105, 15113407062476817, 122714906949538257, 1013127345082389513

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

a(n) is the number of n-step closed walks (from origin to origin) in 4-dimensional lattice where each step changes at most one component by -1 or by +1. - Alois P. Heinz, Oct 26 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1054

Row 4 of A328718.

Cf. A039699.

{a(n) = polcoef(polcoef(polcoef(polcoef((1+w+x+y+z+1/w+1/x+1/y+1/z)^n, 0), 0), 0), 0)}

From Vaclav Kotesovec, Oct 26 2019: (Start)
Recurrence: n^4*a(n) = (5*n^4 - 10*n^3 + 10*n^2 - 5*n + 1)*a(n-1) + (n-1)^2*(70*n^2 - 140*n + 113)*a(n-2) - (n-2)*(n-1)*(230*n^2 - 690*n + 583)*a(n-3) - 789*(n-3)*(n-2)^2*(n-1)*a(n-4) + 945*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 9^(n+2) / (16 * Pi^2 * n^2). (End)
E.g.f.: exp(x) * BesselI(0,2*x)^4. - Ilya Gutkovskiy, Oct 26 2019

a(19)-a(22) from Alois P. Heinz, Oct 26 2019

A356258 Number of 6-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

Original entry on oeis.org

1, 12, 396, 19920, 1281420, 96807312, 8175770064, 748315668672, 72729762868620, 7402621930738320, 781429888276676496, 84955810313787521472, 9463540456205136873936, 1075903653146632508721600, 124461755084172965028753600, 14615050011682746903615601920

Offset: 0

Dave R.M. Langers, Oct 12 2022

			a(1)=12, because twelve paths start at the origin, visit one of the adjacent vertices, and immediately return to the origin, resulting in 12 different paths of length 2n=2*1=2.

Alois P. Heinz, Table of n, a(n) for n = 0..467

Row k=6 of A287318.

1-5 dimensional analogs are A000984, A002894, A002896, A039699, A287317.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> (2*n)!*b(n, 6)/n!^2:
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 30 2023

E.g.f.: Sum_{n>=0} a(2*n) * x^(2*n)/(2*n)! = I_0(2*x)^6. (I = Modified Bessel function first kind).

a(n) = Sum_{h+i+j+k+l+m=n, 0<=h,i,j,k,l,m<=n} multinomial(2n [h,h,i,i,j,j,k,k,l,l,m,m]). - Shel Kaphan, Jan 29 2023

A361364 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

Original entry on oeis.org

1, 10, 170, 6500, 332050, 19784060, 1296395700, 90616189800, 6637652225250, 503852804991500, 39337349077483420, 3142010167321271000, 255747325678297576100, 21150729618673827139000, 1773152567858996728205000, 150409554094012703302602000, 12890454660664800562838261250

Offset: 0

Shel Kaphan, Mar 09 2023

In Novak's note it is mentioned that if P(z) and Q(z) are the g.f.s for the probabilities of indecomposable and decomposable loops, respectively, that P(z) = 1 - 1/Q(z). This works equally well using loop counts rather than probabilities. The g.f.s may be expressed by the series constructed from the sequences of counts of loops of length 2*n. Q(z) for the 5-d case is the series corresponding to A287317.
To satisfy this g.f. equation, a(0) should be 0, but we give it as 1 since there is one trivial loop of 0 steps, and for consistency with related sequences.
To obtain the probability of returning to the point of origin for the first time after 2*n steps, divide a(n) by the total number of walks of length 2*n in d dimensions: (2*d)^(2*n) = 100^n.

Alois P. Heinz, Table of n, a(n) for n = 0..503
Jonathan Novak, Pólya's Random Walk Theorem.

Cf. A287317, A039699 (number of walks that return to the origin in 2n steps).
Number of walks that return to the origin for the first time in 2n steps, in 1..4 dimensions: |A002420|, A054474, A049037, A359801.
Column k=5 of A361397.
Cf. A169714.

walk5d[n_] :=
 Sum[(2 n)!/(i! j! k! l! (n - i - j - k - l)!)^2, {i, 0, n}, {j, 0,
   n - i}, {k, 0, n - i - j}, {l, 0, n - i - j - k}]; invertSeq[seq_] :=
 CoefficientList[1 - 1/SeriesData[x, 0, seq, 0, Length[seq], 1], x]; invertSeq[Table[walk5d[n], {n, 0, 15}]]

G.f.: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^5 dt.

INVERTi transform of A169714.

A198804 Number of closed paths of length n whose steps are 16th roots of unity, U_16(n).

Original entry on oeis.org

1, 0, 16, 0, 720, 0, 50560, 0, 4649680, 0, 514031616, 0, 64941883776, 0, 9071319628800, 0, 1369263687414480, 0, 219705672931613440, 0, 37024402443528248320, 0, 6493814173413849784320, 0, 1177304833671218302960000, 0, 219456611569479963675136000, 0

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

U_16(n), (comment in article) : For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Cf. A000984, A002894, A039699, A126869.

{a(n)=n!*polcoeff(sum(m=0,n,x^(2*m)/m!^2+x*O(x^n))^8,n)}

E.g.f.: ( Sum_{n>=0} x^(2*n)/n!^2 )^8. - Paul D. Hanna, Oct 30 2011

E.g.f.: g(x)^8 where g(x) is the e.g.f. of A126869. - Andrew Howroyd, Nov 01 2018

cp's OEIS Frontend

A385087 2-adic valuation of A039699.

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A287318 Square array A(n,k) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

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A288459 Chebyshev coefficients of density of states of 4D hypercubic lattice.

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A287317 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

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A303503 a(n) = (2*n)! * [x^(2*n)] BesselI(0,2*x)^n.

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A359801 Number of 4-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

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A328714 Constant term in the expansion of (1 + w + x + y + z + 1/w + 1/x + 1/y + 1/z)^n.

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A356258 Number of 6-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

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A287318 Square array A(n,k) = (2n)! [x^n] BesselI(0, 2sqrt(x))^k read by antidiagonals.

A303503 a(n) = (2n)! [x^(2n)] BesselI(0,2x)^n.