A002895 -id:A002895 - cp's OEIS frontend

A242207 Least prime divisor of the n-th Domb number D(n) which does not divide any D(k) with k < n, or 1 if such a primitive prime divisor of D(n) does not exist.

2, 7, 1, 97, 11, 23, 19, 643, 659, 1753, 4922329, 613, 341447, 1193, 2213, 2040452101603, 491, 82461839, 733, 113, 1108394340978316050481, 1034497328556150923437, 593, 73, 17117, 804943, 422291, 1559, 858631, 337655751557

Offset: 1

Zhi-Wei Sun, May 07 2014

nonn

Conjecture: a(n) is prime except for n = 3.

			 a(4) = 97 since D(4) = 2^2*7*97 with 97 dividing none of D(1) = 2^2, D(2) = 2^2*7 and D(3) = 2^8.

Zhi-Wei Sun, Table of n, a(n) for n = 1..72

Cf. A000040, A002895, A242169, A242170, A242171, A242173, A242174, A242193, A242194, A242195.

d[n_]:=Sum[Binomial[n,k]^2*Binomial[2k,k]*Binomial[2(n-k),n-k],{k,0,n}]
f[n_]:=FactorInteger[d[n]]
p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[If[d[n]<2,Goto[cc]];Do[Do[If[Mod[d[i],Part[p[n],k]]==0,Goto[aa]],{i,1,n-1}];Print[n," ",Part[p[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[p[n]]}];Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,30}]

A336639 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^4.

Original entry on oeis.org

1, 4, 36, 544, 12196, 377904, 15438816, 803602944, 51908768676, 4074743122384, 382079412133936, 42184889139337344, 5417567866536188896, 800808722921088352384, 135006904500993157933056, 25751088570167886107910144, 5517695042810314282550432676

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

In general, if k>=1 and Sum_{n>=0} a(n) * x^n / n!^2 = 1 / BesselJ(0, 2*sqrt(x))^k, then a(n) ~ n!^2 * n^(k-1) / ((k-1)! * r^(n + k/2) * BesselJ(1, 2*sqrt(r))^k), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4. - Vaclav Kotesovec, Jul 11 2025

Cf. A000275, A002895, A336271, A336638.

Column k=4 of A340986.

nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^4, {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002895(k) * a(n-k).

a(n) ~ n!^2 * n^3 / (6 * r^(n+2) * BesselJ(1, 2*sqrt(r))^4), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025

A336637 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^4 - 1).

Original entry on oeis.org

1, 4, 60, 1648, 71612, 4448384, 370135632, 39480942848, 5227020747708, 837878205997216, 159457868003008640, 35459969754432262208, 9093585253916177728592, 2659611377508767798535488, 878768601771275773332660736, 325350340926291926090183214848

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

Cf. A002895, A023998, A336635, A336636.

nmax = 15; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^4 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * A002895(k) * k * a(n-k).

A364111 a(n) = Sum_{k = 0..n} binomial(n+k-1,k)^2 * binomial(2n-2k,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 4, 76, 2560, 106060, 4864504, 237354880, 12079462560, 633885607500, 34050190896040, 1863047125801576, 103465470769890112, 5817117095161011328, 330450303019252600240, 18937657945720403830240, 1093557503049551583194560, 63566414131528881235953228, 3716526456851323626808570632

Offset: 0

Peter Bala, Jul 07 2023

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.
More generally, let A >= 2, B and C be positive integers. Then we conjecture that the sequence whose terms are given by Sum_{k = 0..n} binomial(n+k-1,k)^A * binomial(2*n-2*k,n-k)^B * binomial(2*k,k)^C also satisfies the same  supercongruences.

Peter Bala, A recurrence for A364111.
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. A002895, A362676.

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

Table[Binomial[2*n,n] * HypergeometricPFQ[{-n, n, n, 1/2}, {1, 1, 1/2 - n}, 1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 09 2023 *)

a(n) = Sum_{k = 0..n} binomial(-n,k)^2 * binomial(2*n-2*k,n-k) * binomial(2*k,k).
a(n) = binomial(2*n,n)*hypergeom([-n, n, n, 1/2], [1, 1, 1/2 - n], 1).
a(n) ~ 2^(6*n-1) / (sqrt(3) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Jul 09 2023

A174123 Partial sums of A002893.

Original entry on oeis.org

1, 4, 19, 112, 751, 5404, 40573, 313408, 2471167, 19791004, 160459069, 1313922064, 10847561089, 90174127684, 754009158019, 6336733626112, 53489159252671, 453258909448636, 3854034482891725, 32871004555812112, 281127047928811201, 2410285909684370788

Offset: 0

Jonathan Vos Post, Mar 08 2010

Tewodros Amdeberhan and Roberto Tauraso, Two triple binomial sum supercongruences, arXiv:1607.02483 [math.NT], Jul 08 2016.

Cf. A002893, A000172, A002895, A000984, A207321.

Accumulate[Table[Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}]] (* Harvey P. Dale, May 05 2013 *)

a(n)=sum(m=0,n,sum(k=0,m,binomial(m,k)^2*binomial(2*k,k)))

a(n) = Sum_{i=0..n} A002893(i).
From Sergey Perepechko, Feb 16 2011: (Start)
O.g.f.: 2*sqrt(2)/Pi/(1-z)/sqrt(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))* EllipticK(8*z^(3/2)/(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))).
9*(n+2)^2*a(n) - (99+86*n+19*n^2)*a(n+1) + (72+56*n+11*n^2)*a(n+2) - (n+3)^2*a(n+3)=0. (End)
a(n) ~ 3^(2*n + 7/2) / (32*Pi*n). - Vaclav Kotesovec, Jul 11 2016

A329475 a(n) = Sum_{k=0..n} C(n,k)^2T(k)T(n-k), where T(k) = A002426(k) is the coefficient of x^k in the expansion of (x^2+x+1)^k.

Original entry on oeis.org

1, 2, 10, 68, 586, 5252, 49204, 475400, 4723786, 47937812, 494786260, 5177188040, 54794164660, 585565913480, 6309889976680, 68484312535568, 747985368753226, 8214968193003860, 90669516557975524, 1005156080857529768, 11187435500257898836, 124964856185950621832

Offset: 0

Zhi-Wei Sun, Nov 13 2019

nonn

The author introduced this sequence in arXiv:1911.05456 and made the following conjecture.
Conjecture: Let p be an odd prime and let S = Sum_{k=0..p-1}a(k)/(-4)^k. If p == 1 (mod 12) and p = x^2 + 9*y^2 with x and y integers, then  S == 4*x^2-2*p (mod p^2). If p == 5 (mod 12) and p = x^2 + y^2 with x == y (mod 3), then S == 4*x*y (mod p^2). If p == 3 (mod 4), then S == 0 (mod p^2).
Note that if p > 3 is a prime, then a(p-1) == Sum_{k=0..p-1} T(k)*T(p-1-k) == Legendre(p/3)*Sum_{k=0..p-1}T(k)^2/(-3)^k == 1 (mod p) by (1.7) and (2.3) of the author's 2014 paper in Sci. China Math.

			a(1) = 2 since Sum_{k=0,1} C(1,k)^2*T(k)*T(1-k) = C(1,0)^2*T(0)*T(1) + C(1,1)^2*T(1)*T(0) = 2*T(0)*T(1) = 2*1*1 = 2.

Seiichi Manyama, Table of n, a(n) for n = 0..931 (terms 0..150 from Zhi-Wei Sun)
Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57(2014), no.7, 1375-1400.
Zhi-Wei Sun, On sums related to central binomial and trinomial coefficients, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. in Math. & Stat., Vol. 101, Springer, New York, 2014, pp. 257-312. Also available from arXiv:1101.0600 [math.NT], 2011-2014.
Zhi-Wei Sun, New series for powers of Pi and related congruences, Electron. Res. Arch. 28 (2020), no. 3, 1273--1342.
Zhi-Wei Sun, On central trinomial coefficients, Question 491563 at MathOverflow, April 23, 2025.

Cf. A002426, A002895.

T[0]=1; T[1]=1; T[n_]:=T[n]=((2n-1)T[n-1]+3*(n-1)*T[n-2])/n;
a[n_]:=a[n]=Sum[Binomial[n,k]^2*T[k]*T[n-k],{k,0,n}];
Table[a[n],{n,0,21}]

a(n) ~ (3/2)*12^n/(n*Pi)^(3/2) as n tends to the infinity.

A336622 a(n) = Sum_{k=0..n} Sum_{i=0..k} Sum_{j=0..i} (binomial(n,k) * binomial(k,i) * binomial(i,j))^n.

Original entry on oeis.org

1, 4, 28, 1192, 591460, 3441637504, 219272057247376, 185528149944660881488, 2405748000504972140803769860, 349789137657321307953339196885516144, 652520795984468974632890750361094911319873648

Offset: 0

Ilya Gutkovskiy, Jul 29 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..42

Cf. A000302, A002895, A167010, A287699, A336270.

B:=Binomial; [(&+[(&+[(&+[(B(n,j)*B(n-j,k-j)*B(k-j,k-i))^n: j in [0..i]]): i in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022

Table[Sum[Sum[Sum[(Binomial[n, k] Binomial[k, i] Binomial[i, j])^n, {j, 0, i}], {i, 0, k}], {k, 0, n}], {n, 0, 10}]
Table[(n!)^n SeriesCoefficient[Sum[x^k/(k!)^n, {k, 0, n}]^4, {x, 0, n}], {n, 0, 10}]

b=binomial
def A336622(n): return sum(sum(sum( (b(n,j)*b(n-j,k-j)*b(k-j,k-i))^n for j in (0..i)) for i in (0..k)) for k in (0..n))
[A336622(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^4.

A372941 Numbers k that divide the k-th Domb number.

Original entry on oeis.org

1, 2, 4, 14, 28, 112, 133, 176, 224, 368, 388, 448, 616, 704, 784, 896, 1216, 1568, 1792, 3563, 4256, 5144, 6272, 8624, 8924, 9856, 11264, 11776, 13927, 16702, 23408, 32936, 38509, 42238, 43456, 43652, 43904, 46424, 67328, 73784, 76912, 78848, 81466, 110614, 118256

Offset: 1

Amiram Eldar, May 17 2024

nonn

Numbers k such that k | A002895(k).

			2 is a term since A002895(2) = 28 = 2 * 14 is divisible by 2.
4 is a term since A002895(4) = 2716 = 4 * 679 is divisible by 4.

Amiram Eldar, Table of n, a(n) for n = 1..132

Cf. A002895.

Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

seq[kmax_] := Module[{d0 = 1, d1 = 4, d2, s = {1}}, Do[d2 = ((20*k^3 - 30*k^2 + 18*k - 4)*d1 - 64*(k-1)^3*d0)/k^3; If[Divisible[d2, k], AppendTo[s, k]]; d0 = d1; d1 = d2, {k, 2, kmax}]; s]; seq[5000]

lista(kmax) = {my(d0 = 1, d1 = 4, d2); print1("1, "); for(k = 2, kmax, d2 = ((20*k^3 - 30*k^2 + 18*k - 4)*d1 - 64*(k-1)^3*d0)/k^3; if(!(d2 % k), print1(k, ", ")); d0 = d1; d1 = d2);}

A385087 2-adic valuation of A039699.

Original entry on oeis.org

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)).

A385286 a(n) = (n!)^2 [x^n] hypergeom([], [1], x)^8.

Original entry on oeis.org

1, 8, 120, 2528, 66424, 2039808, 70283424, 2643158400, 106391894904, 4518833256512, 200396211454720, 9205443151733760, 435368682010660000, 21100379936684418560, 1044115187294444772480, 52597451834668445910528, 2691037806733052553149304, 139567074682665782246950080

Offset: 0

Peter Luschny, Jun 24 2025

We regard this sequence in the list of sequences n -> A287316(n, 2^k) for k = 3.

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

Cf. A000012 (k=0), A000984 (k=1), A002895 (k=2), this sequence (k=3), A287316.

A385286_list := proc(len) local n; series(hypergeom([], [1], x)^8, x, len);
seq((n!)^2*coeff(%, x, n), n = 0..len-1) end: A385286_list(18);

nmax = 20; CoefficientList[Series[BesselI[0, 2*Sqrt[x]]^8, {x, 0, nmax}], x] * Range[0, nmax]!^2 (* Vaclav Kotesovec, Jun 24 2025 *)

a(n) = my(x='x+O('x^(n+1))); n!^2*polcoeff(hypergeom([], [1], x)^8, n); \\ Michel Marcus, Jun 24 2025

a(n) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^8.
a(n) = A287316(n, 2^3).
a(n) ~ 2^(6*n+5) / (Pi^(7/2) * n^(7/2)). - Vaclav Kotesovec, Jun 24 2025

cp's OEIS Frontend

A242207 Least prime divisor of the n-th Domb number D(n) which does not divide any D(k) with k < n, or 1 if such a primitive prime divisor of D(n) does not exist.

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A336639 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^4.

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A336637 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x))^4 - 1).

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A364111 a(n) = Sum_{k = 0..n} binomial(n+k-1,k)^2 * binomial(2*n-2*k,n-k) * binomial(2*k,k).

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Maple

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A174123 Partial sums of A002893.

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A329475 a(n) = Sum_{k=0..n} C(n,k)^2*T(k)*T(n-k), where T(k) = A002426(k) is the coefficient of x^k in the expansion of (x^2+x+1)^k.

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A336622 a(n) = Sum_{k=0..n} Sum_{i=0..k} Sum_{j=0..i} (binomial(n,k) * binomial(k,i) * binomial(i,j))^n.

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Magma

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A372941 Numbers k that divide the k-th Domb number.

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A364111 a(n) = Sum_{k = 0..n} binomial(n+k-1,k)^2 * binomial(2n-2k,n-k) * binomial(2*k,k).

A329475 a(n) = Sum_{k=0..n} C(n,k)^2T(k)T(n-k), where T(k) = A002426(k) is the coefficient of x^k in the expansion of (x^2+x+1)^k.