A275027 -id:A275027 - cp's OEIS frontend

A349420 Primes that do not divide any term of A275027.

2, 3, 7, 11, 31, 41, 67, 73, 79, 89, 97, 101, 103, 107, 127, 131, 137, 181, 211, 251, 277, 281, 283, 293, 307, 311, 317, 331, 347, 349, 359, 367, 383, 409, 419, 421, 431, 449, 463, 523, 547, 563, 577, 599, 607, 613, 617, 631, 677, 683, 691, 773, 787, 797, 821, 823, 827, 911, 977

Offset: 1

Michel Marcus, Nov 17 2021

nonn

f(n) = A275027(n) is never divisible by a prime p if none of the values f(0), f(1), ..., f(p-1) is divisible by p. See Henningsen and Straub, who ask for an explicit characterization for these primes.

Michel Marcus, Table of n, a(n) for n = 1..289
Joel A. Henningsen and Armin Straub, Generalized Lucas congruences and linear p-schemes, arXiv:2111.08641 [math.NT], 2021.

Cf. A275027.

f[n_] := f[n] = Sum[Binomial[n, k]^2*Binomial[n - k, k], {k, 0, n/2}]; q[p_] := AllTrue[Table[f[k], {k, 2, p - 1}], ! Divisible[#, p] &]; Select[Range[1000], PrimeQ[#] && q[#] &] (* Amiram Eldar, Nov 17 2021 *)

f(n) = sum(k=0, n, binomial(n, k)^2*binomial(n-k, k)); \\ A275027
isdiv(v, n) = {my(p=prime(n)); for (k=1, p, if (!(v[k] % p), return(1));); return(0);}
lista(nn) = {my(p=prime(nn), v=vector(p, k, f(k-1)), list=List()); for(n=1, nn, if (! isdiv(v, n), listput(list, prime(n)););); Vec(list);}

A244973 a(n) = Sum_{k=0..n} (-1)^kC(n, k)^2C(2k, k), where C(n, k) denotes the binomial coefficient n!/(k!(n-k)!).

Original entry on oeis.org

1, -1, -1, 17, -65, 49, 881, -5489, 12223, 42785, -479951, 1746271, 440881, -39651457, 212039855, -326783183, -2817155137, 23175692033, -68726927071, -128775914225, 2285692892785, -10156877725985, 6169206210815, 196882990135745, -1274770281690575

Offset: 0

Zhi-Wei Sun, Jul 08 2014

Zhi-Wei Sun introduced this sequence in arXiv:1407.0967. For any prime p > 5, he proved that Sum_{k=1..p-1} a(k)/k^2 == 0 (mod p) and Sum_{k=1..p-1} a(k)/k == 0 (mod p^2). This is quite similar to Wolstenholme's congruences Sum_{k=1..p-1} 1/k^2 == 0 (mod p) and Sum_{k=1..p-1} 1/k == 0 (mod p^2) for any prime p > 3.
Conjecture: For any prime p > 5 and positive integer n, the number (a(p*n) - a(n))/(p*n)^3 is always a p-adic integer.
The author proved a weaker version of this in arXiv:1610.03384. - Zhi-Wei Sun, Nov 12 2016

			a(3) = 17 since C(3,0)^2*C(2*0,0) - C(3,1)^2*C(2,1) + C(3,2)^2*C(4,2) - C(3,3)^2*C(6,3) = 1 - 18 + 54 - 20 = 17.

Zhi-Wei Sun, Table of n, a(n) for n = 0..200
V. J. Guo, G.-S. Mao and H. Pan, Proof of a conjecture involving Sun polynomials, J. Difference Equ. Appl., 22(2016), no. 8, 1184-1197; also arXiv:1511.04005 [math.NT], 2015.
Zhi-Wei Sun, Congruences involving g_n(x) = sum_{k=0}^n C(n,k)^2*C(2k,k)*x^k, arXiv:1407.0967 [math.NT], 2014-2016.
Zhi-Wei Sun, Congruences involving g_n(x) = sum_{k=0}^n C(n,k)^2*C(2k,k)*x^k, Ramanujan J. 40 (2016), no. 3, 511-533.
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1016.03384 [math.NT], 2016.

Cf. A000172, A002893, A275027, A277640.

a := n -> hypergeom([1/2, -n, -n], [1, 1], -4):
seq(simplify(a(n)), n = 0..24);  # Peter Luschny, Mar 16 2025

s[n_]:=Sum[Binomial[n,k]^2*Binomial[2k,k](-1)^k,{k,0,n}]
Table[s[n],{n,0,20}]

a(n) = sum(k=0, n, (-1)^k*binomial(n,k)^2*binomial(2*k,k)); \\ Michel Marcus, Nov 13 2016

Recurrence (obtained via the Zeilberger algorithm):
(n+3)^2*(4n+5)*a(n+3) + (20n^3+125n^2+254n+165)*a(n+2) + (76n^3+399n^2+678n+375)*a(n+1) - 25*(n+1)^2*(4n+9)*a(n) = 0.
Lim_sup_{n->oo} |a(n)|^(1/n) = 5. - Vaclav Kotesovec, Jul 13 2014
a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,2*k)^2*C(2*k,k) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(n,2*k)*C(n-k,k). - Zhi-Wei Sun, Nov 12 2016
Conjecture: a(n) = Sum_{k = 0..n} binomial(n, k)*b(k), where b(n) = Sum_{k = 0..n} (-1)^k*binomial(n, k)^2*binomial(2*k, n). [Added Mar 16 2025: this conjecture can be verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package for Maple]. - Peter Bala, Jul 19 2024

A278405 a(n) = Sum_{k=0..n} binomial(n,2k)^2*binomial(n-k,k).

Original entry on oeis.org

1, 1, 2, 19, 110, 476, 2477, 15093, 86830, 485290, 2826902, 16857116, 100034453, 594833357, 3574477090, 21611465819, 130955824174, 796195223398, 4860425688176, 29760574848750, 182655048136510, 1123720751229858, 6929124085148938, 42811398244528788

Offset: 0

Zhi-Wei Sun, Nov 20 2016

nonn

Conjecture: For any prime p > 5 and positive integer n, the number (a(p*n)-a(n))/(p*n)^3 is always a p-adic integer.
We have proved that for any prime p > 5 and positive integer n the number (a(p*n)-a(n))/(p^3*n^2) is always a p-adic integer.
Diagonal of the rational function 1 / ((1 + x)*(1 - x)*(1 - y)*(1 - z) - x*y*z). - Ilya Gutkovskiy, Apr 23 2025

			a(3) = 19 since a(3) = C(3,2*0)^2*C(3-0,0) + C(3,2*1)^2*C(3-1,1) = 1 + 3^2*2 = 19.
G.f. = 1 + x + 2*x^2 + 19*x^3 + 110*x^4 + 476*x^5 + 2477*x^6 + 15093*x^7 + ...

Zhi-Wei Sun, Table of n, a(n) for n = 0..200
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A208425, A244973, A275027, A277640, A278415.

a[n_]:=a[n]=Sum[Binomial[n,2k]^2*Binomial[n-k,k],{k,0,n/2}]
Table[a[n],{n,0,27}]

A278415 a(n) = Sum_{k=0..n} binomial(n, 2k)binomial(n-k, k)(-1)^k.

Original entry on oeis.org

1, 1, 0, -5, -16, -24, 15, 197, 576, 724, -1200, -8832, -22801, -21293, 76440, 408795, 922368, 499104, -4446588, -19025060, -37012416, -1673992, 245604832, 880263936, 1441226991, -908700649, -13088509200, -40222012703, -52991533744, 88167061704, 678172355415, 1805175708261, 1747974632448, -6237554623536, -34300087628480

Offset: 0

Zhi-Wei Sun, Nov 21 2016

sign

Conjecture: For any prime p > 3 and positive integer n, the number (a(p*n)-a(n))/(p*n)^2 is always a p-adic integer.
We are able to show that for any prime p > 3 and positive integer n the number (a(p*n)-a(n))/(p^2*n) is always a p-adic integer.
See also A275027 and A278405 for similar conjectures.

			a(3) = -5 since a(3) = C(3, 2*0)*C(3-0, 0)(-1)^0 + C(3,2*1)*C(3-1,1)(-1)^1 = 1 - 6 = -5.

Zhi-Wei Sun, Table of n, a(n) for n = 0..400
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A208425, A244973, A277640, A275027, A278405.

a[n_]:=Sum[Binomial[n,2k]Binomial[n-k,k](-1)^k,{k,0,n}]
Table[a[n],{n,0,34}]

A382841 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^2.

Original entry on oeis.org

1, 1, 5, 37, 181, 1301, 9401, 65465, 498037, 3796021, 29221705, 230396585, 1828448425, 14651160265, 118544522045, 965075143037, 7907605360757, 65162569952245, 539515760866889, 4486877961224297, 37463151704756281, 313909383754331801, 2638892573249746445, 22249830926517611917

Offset: 0

Ilya Gutkovskiy, Apr 06 2025

nonn

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y)^2*z*w).

Alois P. Heinz, Table of n, a(n) for n = 0..1053

Cf. A000984, A002426, A005259, A005260, A051286, A181546, A275027, A277247, A382842.

Cf. A089627.

a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^2, k=0..n/2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 07 2025

Table[Sum[(Binomial[n, k] Binomial[n - k, k])^2, {k, 0, Floor[n/2]}], {n, 0, 23}]
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1, 1}, 16], {n, 0, 23}]
Table[SeriesCoefficient[1/((1 - x) (1 - y) (1 - z) (1 - w) - (x y)^2 z w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {w, 0, n}], {n, 0, 23}]

a(n) ~ 3^(2*n+2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 07 2025

a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^2. - Alois P. Heinz, Apr 07 2025

A381482 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)^2 * binomial(n-k,k) * 2^k.

Original entry on oeis.org

1, 1, 9, 37, 241, 1401, 8961, 57429, 377217, 2509201, 16876729, 114600069, 783903121, 5397915433, 37372017489, 259998843477, 1816376953857, 12736545070113, 89602978644969, 632223913939557, 4472680961409201, 31717890254271321, 225416254500886689, 1605197563027768917

Offset: 0

Ilya Gutkovskiy, Apr 22 2025

nonn

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z) - 2*x^2*y*z).

Cf. A001045, A001850, A084601, A206178, A275027.

Table[Sum[Binomial[n, k]^2 Binomial[n - k, k] 2^k, {k, 0, Floor[n/2]}], {n, 0, 23}]
Table[HypergeometricPFQ[{1/2 - n/2, -n, -n/2}, {1, 1}, -8], {n, 0, 23}]

a(n) = hypergeom( [1/2 - n/2, -n, -n/2], [1, 1], -8).

a(n) ~ sqrt(7/12 + sqrt(89/38)*cosh(arccosh((8567*sqrt(19/178))/1424)/3)/3) * ((1/3 + 8*sqrt(7)*(cosh(arccosh(1261/(448*sqrt(7)))/3)/3))^n / Pi) / n. - Vaclav Kotesovec, Apr 23 2025

A383524 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n-k,k)^2.

Original entry on oeis.org

1, 1, 3, 13, 43, 171, 711, 2913, 12363, 53203, 230593, 1010703, 4463119, 19827679, 88594299, 397741893, 1793063883, 8113429419, 36832823289, 167701920759, 765577205433, 3503296744233, 16065995216109, 73824301464939, 339844364816559, 1567063753104471, 7237078197034221

Offset: 0

Seiichi Manyama, Apr 29 2025

Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - x*y^2*z^2).

Cf. A000172, A181545, A275027.

Table[Sum[Binomial[n,k] * Binomial[n-k,k]^2,{k,0,n}], {n,0,30}] (* Vaclav Kotesovec, Aug 30 2025 *)

a(n) = sum(k=0, n\2, binomial(n, k)*binomial(n-k, k)^2);

cp's OEIS Frontend

A349420 Primes that do not divide any term of A275027.

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A244973 a(n) = Sum_{k=0..n} (-1)^k*C(n, k)^2*C(2*k, k), where C(n, k) denotes the binomial coefficient n!/(k!*(n-k)!).

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A278405 a(n) = Sum_{k=0..n} binomial(n,2k)^2*binomial(n-k,k).

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A278415 a(n) = Sum_{k=0..n} binomial(n, 2k)*binomial(n-k, k)*(-1)^k.

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A382841 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^2.

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A381482 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)^2 * binomial(n-k,k) * 2^k.

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A383524 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n-k,k)^2.

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A244973 a(n) = Sum_{k=0..n} (-1)^kC(n, k)^2C(2k, k), where C(n, k) denotes the binomial coefficient n!/(k!(n-k)!).

A278415 a(n) = Sum_{k=0..n} binomial(n, 2k)binomial(n-k, k)(-1)^k.