A178790 -id:A178790 - cp's OEIS frontend

A178791 The arithmetic mean of (2k+1)(-1)^k*A_k (k=0,...,n-1), where A_0, A_1,... are Apéry numbers given by A005259.

1, -7, 117, -2441, 57449, -1453635, 38609845, -1061792695, 29973352185, -863536596143, 25288254409373, -750531594051981, 22525211241191881, -682459907754004723, 20845409947239778533, -641211780685502724425

Offset: 1

Zhi-Wei Sun, Jun 14 2010

sign

On Jun 14 2010, Zhi-Wei Sun conjectured that the number a(n) = n^{-1}*Sum_{k=0..n-1} (2*k+1)*(-1)^k*A_k is always an integer and that a(p) = p(p/3) (mod p^3) for any prime p>3. He also formulated the following conjecture: If p=1 (mod 3) is a prime and p = x^2 + 3y^2 with x,y integers then Sum_{k=0..p-1} (-1)^k*A_k = 4*x^2 - 2*p (mod p^2); if p=2 (mod 3) is a prime then Sum_{k=0..p-1 }(-1)^k*A_k=0 (mod p^2).

			For n=3 we have a(3) = (A_0 - 3A_1 + 5A_2)/3 = (1 - 3*5 + 5*73)/3 = 117.

G. C. Greubel, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, arXiv:1006.2776 [math.NT], 2010-2011.

Cf. A178790, A005259, A173774.

G := (-1/2)*(3*x-3+(x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3, 2/3], [1], (-1/2)*(x^2-7*x+1)*(x+1)^(-3)*(x^2-34*x+1)^(1/2)+(1/2)*(x^3+30*x^2 -24*x+1)*(x+1)^(-3))^2;
ogf := 2*x*G/(x+1)+Int((x-1)*G/(x+1)^2,x);
series(ogf, x=0, 25);
series(-subs(x=-x,%), x=0, 25); # Mark van Hoeij, May 07 2013

Apery[n_]:= Sum[Binomial[n+k,k]^2Binomial[n,k]^2,{k,0,n}]; AA[n_]:= Sum[(2k+1)(-1)^k*Apery[k],{k,0,n-1}]/n; Table[AA[n],{n,25}]

A(n) = sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2); \\ A005259
a(n) = sum(k=0, n-1, (2*k+1)*(-1)^k*A(k))/n; \\ Michel Marcus, Jan 24 2019

G.f.: apart from the minus signs (just replace x by -x) the generating function is 2*x*G/(x+1) + Int((x-1)*G/(x+1)^2, x) where G is the generating function of A005259. - Mark van Hoeij, May 07 2013

a(n) ~ -(-1)^n * 2^(3/4) * (1 + sqrt(2))^(4*n) / (24 * (Pi*n)^(3/2)). - Vaclav Kotesovec, Jan 24 2019

A178808 a(n) = (1/n^2) * Sum_{k = 0..n-1} (2k+1)(D_k)^2, where D_0, D_1, ... are central Delannoy numbers given by A001850.

Original entry on oeis.org

1, 7, 97, 1791, 38241, 892039, 22092673, 571387903, 15271248769, 418796912007, 11725812711009, 333962374092543, 9648543623050593, 282164539499639559, 8338391167566634497, 248661515283002490879, 7474768663941435203073

Offset: 1

Zhi-Wei Sun, Jun 16 2010

nonn

On Jun 14 2010, Zhi-Wei Sun conjectured that a(n) = (1/n^2) * Sum_{k = 0..n-1} (2*k+1)*(D_k)^2 is always an integer and that p^2*a(p) = p^2 - 4*p^3*q_p(2) - 2*p^4*q_p(2)^2 (mod p^5) for any prime p > 3, where q_p(2) denotes the Fermat quotient (2^(p-1) - 1)/p (see Sun, Remark 4.3, p. 26, 2014). He also conjectured that Sum_{k = 0..n-1} (2*k+1)*(-1)^k*(D_k)^2 == 0 (mod n*D_n/(3,D_n)) for all n = 1,2,3,....

The fact that a(n) is an integer follows directly from the formulas for a(n) in the formula section below. - Mark van Hoeij, Nov 13 2022

			For n = 3 we have a(3) = (D_0^2 + 3*D_1^2 + 5*D_2^2)/3^2 = (1 + 3*3^2 + 5*13^2)/3^2 = 97.

G. C. Greubel, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, arXiv:1006.2776 [math.NT], 2011.
Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57 (2014), no. 7, 1375-1400; arXiv:1008.3887 [math.NT], 2010-2014.

Cf. A001850, A178790, A178791, A173774, A358388.

A001850 := n -> LegendreP(n, 3); seq((6*A001850(n)*A001850(n-1)-A001850(n)^2-A001850(n-1)^2)/8, n=1..20); # Mark van Hoeij, Nov 12 2022
# Alternative:
g := n -> hypergeom([n, -n, 1/2], [1, 1], -8): # A358388
f := n -> hypergeom([-n, -n], [1], 2):         # A001850
a := n -> (3*f(n)*f(n-1) - g(n)) / 4:
seq(simplify(a(n)), n = 1..17); # Peter Luschny, Nov 13 2022

DD[n_]:=Sum[Binomial[n+k,2k]Binomial[2k,k],{k,0,n}]; SS[n_]:= Sum[(2k+1)*DD[k]^2,{k,0,n-1}]/n^2; Table[SS[n],{n,1,25}]
Table[Sum[(2k+1)*JacobiP[k,0,0,3]^2, {k, 0, n-1}]/n^2, {n, 1, 30}] (* G. C. Greubel, Jan 23 2019 *)

# prepends a(0) = 0
def A178808List(size: int) -> list[int]:
    A358387 = A358387gen()
    A358388 = A358388gen()
    return [(next(A358387) - next(A358388)) // 4 for n in range(size)]
print(A178808List(18)) # Peter Luschny, Nov 15 2022

a(n) ~ (1 + sqrt(2))^(4*n) / (16*Pi*n^2). - Vaclav Kotesovec, Jan 24 2019
G.f.: Integral(hypergeom([1/2, 1/2], [2], -32*x/(1 - 34*x + x^2))/((1 - x)*(1 - 34*x + x^2)^(1/2))). - Mark van Hoeij, Nov 10 2022
a(n) = (6*A001850(n)*A001850(n-1) - A001850(n)^2 - A001850(n-1)^2)/8. - Mark van Hoeij, Nov 12 2022
a(n) = (3*f(n)*f(n-1) - g(n))/4, where g(n) = hypergeom([n, -n, 1/2], [1, 1], -8) and f(n) = hypergeom([-n, -n], [1], 2). This formula also gives an integer value for n = 0. - Peter Luschny, Nov 13 2022

A189766 Trace of the inverse of the n-th order Hilbert matrix.

Original entry on oeis.org

1, 16, 381, 10496, 307505, 9316560, 288307285, 9052917760, 287307428985, 9192433560080, 295998598024613, 9580548525151488, 311414673789269713, 10158681128480830288, 332394269045633574405, 10904463909222273843200, 358543696456299951516425

Offset: 1

T. D. Noe, May 02 2011

nonn

See the Mathematica program for a formula in terms of a hypergeometric function.

Vincenzo Librandi, Table of n, a(n) for n = 1..600

Cf. A005249 (determinant), A189765 (inverse Hilbert matrix).

Table[Trace[Inverse[HilbertMatrix[n]]], {n, 20}] (* or *)
Table[n^2 HypergeometricPFQ[{1/2, 1-n, 1-n, 1+n, 1+n}, {1, 1, 1, 3/2}, 1], {n, 20}]

a(n) = trace(1/mathilbert(n)) \\ Jianing Song, Oct 18 2021

From Richard Penner, Jun 05 2011: (Start)
a(n) = n * A178790(n) = Sum_{k=0..n-1} (2*k+1)*binomial(n+k, 2*k+1)^2 * binomial(2*k,k)^2.
a(n) = Sum_{k=1..n} A005408(k)*A005259(k-1) = Sum_{k=0..n-1} (2*k+1) * Sum_{j=0..k} binomial(k+j,j)^2 * binomial(k,j)^2. (End)
Recurrence: (n-1)^3*(2*n-5)*a(n) = (2*n-5)*(35*n^3 - 122*n^2 + 132*n - 40)*a(n-1) - (2*n-1)*(35*n^3 - 193*n^2 + 345*n - 203)*a(n-2) + (n-2)^3*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ 2^(1/4)*(17+12*sqrt(2))^n/(16*Pi^(3/2)*sqrt(n)). - Vaclav Kotesovec, Aug 18 2013

A179089 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)*T_k^2(-3)^(n-1-k), where T_0, T_1, ... are central trinomial coefficients given by A002426.

Original entry on oeis.org

1, 0, 5, 13, 105, 576, 4005, 27000, 193193, 1402672, 10433709, 78807785, 603996745, 4683970032, 36702939429, 290184446349, 2312460578025, 18556825469040, 149842592021997, 1216719520281045, 9929612901775761, 81406058258856240

Offset: 1

Zhi-Wei Sun, Jun 29 2010

nonn

On Jun 28 2010, Zhi-Wei Sun conjectured that a(n) is an integer for every n=1,2,3,... and that a(p) = (1+p/3)/2 (mod p) for any prime p, where (p/3) is the Legendre symbol. In contrast, he showed that Sum_{k=0..n-1} (2k+1)*T_k^2*3^(n-1-k) = n*T_n*T_{n-1} for all n=1,2,3,...

The formula for a(n) in the formula section implies that a(n) is an integer. - Mark van Hoeij, Nov 13 2022

			For n = 4 we have a(4) = (T_0^2(-3)^3 + 3*T_1^2(-3)^2 + 5*T_2^2(-3) + 7*T_3^2)/4^2 = (-27 + 27 - 5*27 + 7^3)/16 = 13.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Cf. A002426, A178808, A178790, A178791, A173774.

A002426 := n -> simplify(GegenbauerC(n, -n, -1/2)); seq( (A002426(n)+A002426(n-1))*(3*A002426(n-1)-A002426(n))/4, n=1..20); # Mark van Hoeij, Nov 13 2022

TT[n_]:=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,Floor[n/2]}] SS[n_]:=Sum[(2k+1)*TT[k]^2*(-3)^(n-1-k),{k,0,n-1}]/n^2 Table[SS[n],{n,1,50}]

G.f.: Integral(hypergeom([1/2, 3/2], [2], 16*x/(1 + 3*x)^2)/(1 + 3*x)^2). - Mark van Hoeij, Nov 10 2022

a(n) = (A002426(n)+A002426(n-1))*(3*A002426(n-1)-A002426(n))/4. - Mark van Hoeij, Nov 13 2022

A179524 a(n) = Sum_{k=0..n} (-4)^kbinomial(n,k)^2binomial(n-k,k)^2.

Original entry on oeis.org

1, 1, -15, -143, 1, 12801, 100401, -555855, -16006143, -69903359, 1371541105, 20881151985, 5878439425, -2725373454335, -25310084063055, 145439041081137, 4851621446905857, 23952290336559105, -470461357757965071, -7793050905481342863, -4149447893184517119

Offset: 0

Zhi-Wei Sun, Jul 17 2010

sign

On July 1, 2010 Zhi-Wei Sun introduced this sequence and made the following conjecture: If p is a prime with p=1,9 (mod 20) and p=x^2+5y^2 with x,y integers, then sum_{k=0}^{p-1}a(k)=4x^2-2p (mod p^2); if p is a prime with p=3,7 (mod 20) and 2p=x^2+5y^2 with x,y integers, then sum_{k=0}^{p-1}a(k)=2x^2-2p (mod p^2); if p is a prime with p=11,13,17,19 (mod 20), then sum_{k=0}^{p-1}w_k=0 (mod p^2). He also conjectured that sum_{k=0}^{n-1}(20k+17)w_k=0 (mod n) for all n=1,2,3,... and that sum_{k=0}^{p-1}(20k+17)w_k=p(10(-1/p)+7) (mod p^2) for any odd prime p. Sun also formulated similar conjectures for some sequences similar to a(n).

			For n=3 we have a(3)=1-4*3^2*2^2=-143.

Zhi-Wei Sun, Open Conjectures on Congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2011.
Zhi-Wei Sun, On Apery numbers and generalized central trinomial coefficients, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Cf. A005259, A178790, A178791, A178808, A179508, A173774.

W[n_]:=Sum[(-4)^k*Binomial[n,k]^2*Binomial[n-k,k]^2,{k,0,n}] Table[W[n],{n,0,50}]

a(n) = Sum_{k=0..[n/2]} (-4)^k*binomial(n,2k)^2*binomial(2k,k)^2.

A179535 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n-k,k)^2 * 81^k.

Original entry on oeis.org

1, 1, 325, 2917, 247861, 5937301, 265793401, 10705726585, 378746444917, 18588932910901, 657940881863305, 32580334626782185, 1257522211980656425, 59212895251349313865, 2490039488311462939645, 112553667120196462181437

Offset: 0

Zhi-Wei Sun, Jul 18 2010

nonn

On Jul 17 2010, Zhi-Wei Sun introduced this sequence and made the following conjecture: If p is a prime with p=1,9,11,19 (mod 40) and p = x^2+10y^2 with x,y integers, then Sum_{k=0..p-1} a(k) == 4x^2-2p (mod p^2); if p is a prime with p == 7,13,23,37 (mod 40) and 2p = x^2 + 10y^2 with x,y integers, then Sum_{k=0..p-1} a(k) == 2p - 2x^2 (mod p^2); if p is an odd prime with (-10/p)=-1, then Sum_{k=0..p-1} a(k) == 0 (mod p^2). He also conjectured that Sum_{k=0..n-1} (10k+9)*a(k) == 0 (mod n) for all n=1,2,3,... and that Sum_{k=0..p-1} (10k+9)*a(k) == p(4(-2/p)+5) (mod p^2) for any prime p > 3.

			For n=2 we have a(2) = 1 + 2^2*81 = 325.

Zhi-Wei Sun, Open Conjectures on Congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2011.
Zhi-Wei Sun, On Apery numbers and generalized central trinomial coefficients, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Cf. A179524, A178790, A178791, A178808, A173774.

a[n_]:=Sum[Binomial[n,k]^2Binomial[n-k,k]^2*81^k,{k,0,n}] Table[a[n],{n,0,25}]

A179100 a(n) = (1/n) * Sum_{k=0..n-1} (8k+5) T_k^2, where T_0, T_1, ... are central trinomial coefficients given by A002426.

Original entry on oeis.org

5, 9, 69, 407, 2997, 22005, 169389, 1325889, 10573677, 85386881, 697013325, 5739021051, 47599593941, 397234035333, 3332690347437, 28089543969855, 237711099004461, 2018856328439841, 17200553934626253, 146966002696538271

Offset: 1

Zhi-Wei Sun, Jun 29 2010

nonn

On Jun 17 2010, Zhi-Wei Sun conjectured that a(n) is an integer for every n=1,2,3,... and that a(p) == 3(p/3) (mod p) for any prime p, where (p/3) is the Legendre symbol. He also observed that Sum_{k=0..n-1} (2k+1) T_k*3^{n-1-k} = n * Sum_{k=0..n-1} C(n-1,k)*(-1)^(n-1-k)*(k+1)*C(2k,k).

			For n=3 we have a(3) = (5*T_0^2 + 13*T_1^2 + 21*T_2^2)/3 = (5 + 13 + 21*9)/3 = 69.

Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Cf. A002426, A179089, A178808, A178790, A178791, A173774.

TT[n_]:=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,Floor[n/2]}] SS[n_]:=Sum[(8k+5)*TT[k]^2,{k,0,n-1}]/n Table[SS[n],{n,1,50}]

A179537 a(n) = Sum_{k=0..n} binomial(n,k)^2binomial(n-k,k)^2(-16)^k.

Original entry on oeis.org

1, 1, -63, -575, 6913, 224001, 420801, -69020223, -918270975, 14596918273, 511845045697, 336721812417, -198449271643391, -2498857696947455, 51614254703660481, 1666776235855331265, -1588877076116525055

Offset: 0

Zhi-Wei Sun, Jul 18 2010

sign

On July 17, 2010 Zhi-Wei Sun introduced this sequence and made the following conjecture: If p is a prime with (p/7)=1 and p=x^2+7y^2 with x,y integers, then sum_{k=0}^{p-1}(-1)^k*a(k)=4x^2-2p (mod p^2); if p is a prime with (p/7)=-1, then sum_{k=0}^{p-1}(-1)^k*a(k)=0 (mod p^2). He also conjectured that sum_{k=0}^{n-1}(42k+37)(-1)^k*a(k)=0 (mod n) for all n=1,2,3,... and that sum_{k=0}^{p-1}(42k+37)(-1)^k*a(k)=p(21(p/7)+16) (mod p^2) for any prime p.

			For n=2 we have a(2)=1+2^2*(-16)=-63.

Zhi-Wei Sun, Open Conjectures on Congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2011.
Zhi-Wei Sun, On Apery numbers and generalized central trinomial coefficients, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Cf. A179536, A179535, A179524, A178790, A178791, A178808, A173774.

a[n_]:=Sum[Binomial[n,k]^2Binomial[n-k,k]^2*(-16)^k,{k,0,n}] Table[a[n],{n,0,25}]

cp's OEIS Frontend

A178791 The arithmetic mean of (2k+1)(-1)^k*A_k (k=0,...,n-1), where A_0, A_1,... are Apéry numbers given by A005259.

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A178808 a(n) = (1/n^2) * Sum_{k = 0..n-1} (2*k+1)*(D_k)^2, where D_0, D_1, ... are central Delannoy numbers given by A001850.

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A189766 Trace of the inverse of the n-th order Hilbert matrix.

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A179089 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)*T_k^2(-3)^(n-1-k), where T_0, T_1, ... are central trinomial coefficients given by A002426.

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A179524 a(n) = Sum_{k=0..n} (-4)^k*binomial(n,k)^2*binomial(n-k,k)^2.

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A179535 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n-k,k)^2 * 81^k.

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A179100 a(n) = (1/n) * Sum_{k=0..n-1} (8k+5) T_k^2, where T_0, T_1, ... are central trinomial coefficients given by A002426.

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A178808 a(n) = (1/n^2) * Sum_{k = 0..n-1} (2k+1)(D_k)^2, where D_0, D_1, ... are central Delannoy numbers given by A001850.

A179524 a(n) = Sum_{k=0..n} (-4)^kbinomial(n,k)^2binomial(n-k,k)^2.

A179537 a(n) = Sum_{k=0..n} binomial(n,k)^2binomial(n-k,k)^2(-16)^k.