A363864 -id:A363864 - cp's OEIS frontend

A363871 a(n) = A108625(2n, 3n).

1, 37, 5321, 980407, 201186025, 43859522037, 9939874413899, 2314357836947571, 549694303511409641, 132569070434503802605, 32360243622138480889321, 7977001183875449759759807, 1982402220908671654519130731, 496031095735572731850517509727

Offset: 0

Peter Bala, Jun 27 2023

a(n) = B(2*n, 3*n, 2*n) in the notation of Straub, equation 24. It follows from Straub, Theorem 3.2, that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
For other cases, see A099601 (r = 2, s = 1), A363867 (r = 1, s = 2), A363868 (r = 3, s = 1), A363869 (r = 3, s = 2) and A363870 (r = 1, s = 3).

G. C. Greubel, Table of n, a(n) for n = 0..400
Peter Bala, A recurrence for A363871

Cf. A005258, A099601, A108625, A363864 - A363870.

A363871:= func< n | (&+[Binomial(2*n,j)^2*Binomial(5*n-j,2*n): j in [0..2*n]]) >;
[A363871(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):
seq(simplify(A108625(2*n, 3*n)), n = 0..20);

Table[HypergeometricPFQ[{-2*n,-3*n,2*n+1}, {1,1}, 1], {n,0,30}] (* G. C. Greubel, Oct 05 2023 *)

def A363871(n): return sum(binomial(2*n,j)^2*binomial(5*n-j,2*n) for j in range(2*n+1))
[A363871(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

a(n) = Sum_{k = 0..2*n} binomial(2*n, k)^2 * binomial(5*n-k, 2*n).
a(n) = Sum_{k = 0..2*n} (-1)^k * binomial(2*n, k)*binomial(5*n-k, 2*n)^2.
a(n) = hypergeometric3F2([-2*n, -3*n, 2*n+1], [1, 1], 1).
a(n) = [x^(3*n)] 1/(1 - x)*Legendre_P(2*n, (1 + x)/(1 - x)).
a(n) ~ sqrt(1700 + 530*sqrt(10)) * (98729 + 31220*sqrt(10))^n / (120 * Pi * n * 3^(6*n)). - Vaclav Kotesovec, Feb 17 2024
a(n) = Sum_{k = 0..2*n} binomial(2*n, k) * binomial(3*n, k) * binomial(2*n+k, k). - Peter Bala, Feb 26 2024

A363867 a(n) = A108625(n,2*n).

Original entry on oeis.org

1, 5, 61, 923, 15421, 272755, 5006275, 94307855, 1811113021, 35301145037, 696227550811, 13863654392945, 278264498108611, 5622746346645953, 114268249446672151, 2333733620675302423, 47868774493665731645, 985608360056821004233, 20362035153323824192645

Offset: 0

Peter Bala, Jun 27 2023

a(n) = B(n,2*n,n) in the notation of Straub, equation 24. It follows from Straub, Theorem 3.2, that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
For other cases, see A099601 (r = 2, s = 1), A363868 (r = 3, s = 1), A363869 (r = 3, s = 2), A363870 (r = 1, s = 3) and A363871 (r = 2, s = 3).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A005258, A099601, A108625, A363864 - A363871.

A363867:= func< n | (&+[Binomial(n,j)^2*Binomial(2*n+j,n): j in [0..n]]) >;
[A363867(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):
seq(simplify(A108625(n, 2*n)), n = 0..18);

Table[HypergeometricPFQ[{-n,-2*n,n+1}, {1,1}, 1], {n,0,30}] (* G. C. Greubel, Oct 05 2023 *)

def A363867(n): return sum(binomial(n,j)^2*binomial(2*n+j,n) for j in range(n+1))
[A363867(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

a(n) = Sum_{k = 0..n} binomial(n, k)^2 * binomial(2*n+k, n).
a(n) = Sum_{k = 0..n} (-1)^(n+k)* binomial(n, k)*binomial(2*n+k, n)^2.
a(n) = hypergeom( [-n, -2*n, n+1], [1, 1], 1).
a(n) = [x^(2*n)] 1/(1 - x)*Legendre_P(n, (1 + x)/(1 - x)).
P-recursive: 4*(2*n - 1)^2*n^2*(85*n^2 - 235*n + 163)*a(n) = (29665*n^6 - 141345*n^5 + 264772*n^4 - 249181*n^3 + 124975*n^2 - 31902*n + 3276)*a(n-1) + 4*(2*n - 3)^2*(n-1)^2*(85*n^2 - 65*n + 13)*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k, k)*binomial(2*n, k). - Peter Bala, Feb 25 2024
a(n) ~ sqrt(13 + 53/sqrt(17)) * (349 + 85*sqrt(17))^n / (Pi * n * 2^(5*n + 5/2)). - Vaclav Kotesovec, Apr 26 2024

A363868 a(n) = A108625(3*n, n).

Original entry on oeis.org

1, 13, 505, 24691, 1337961, 76869013, 4586370139, 280973874215, 17552736006121, 1113134497824901, 71437216036404505, 4629194489296980715, 302391678415222922475, 19886936616891022422159, 1315438146193644502479255, 87445220568000089973356191, 5838332204000163260729138153

Offset: 0

Peter Bala, Jun 27 2023

a(n) = B(3*n, n, 3*n) in the notation of Straub, equation 24. It follows from Straub, Theorem 3.2, that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
For other cases, see A099601 (r = 2, s = 1), A363867 (r = 1, s = 2), A363869 (r = 3, s = 2), A363870 (r = 1, s = 3) and A363871(r = 2, s = 3).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A005258, A099601, A108625, A363864 - A363871.

A363868:= func< n | (&+[Binomial(3*n,n-j)^2*Binomial(3*n+j,j): j in [0..n]]) >;
[A363868(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):
seq(simplify(A108625(3*n, n)), n = 0..16);

Table[HypergeometricPFQ[{-3*n,-n,3*n+1}, {1,1}, 1], {n,0,30}] (* G. C. Greubel, Oct 05 2023 *)

def A363868(n): return sum(binomial(3*n,n-j)^2*binomial(3*n+j, j) for j in range(n+1))
[A363868(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

a(n) = Sum_{k = 0..n} binomial(3*n, n-k)^2 * binomial(3*n+k, k).
a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(3*n, n-k) * binomial(3*n+k, k)^2.
a(n) = hypergeometric3F2( [-3*n, -n, 3*n+1], [1, 1], 1).
a(n) = [x^n] 1/(1 - x)*Legendre_P(3*n, (1 + x)/(1 - x)).
P-recursive: 3*(4797*n^4 - 26076*n^3 + 53055*n^2 - 47886*n + 16178)*(3*n - 1)^2*(3*n - 2)^2*n^2*a(n) = (82935333*n^10 - 699633963*n^9 + 2570641767*n^8 - 5402404662*n^7 + 7171181427*n^6 - 6264762171*n^5 + 3637752517*n^4 - 1382756780*n^3 + 328531700*n^2 - 44004160*n + 2529600)*a(n-1) + 3*(4797*n^4 - 6888*n^3 + 3609*n^2 - 816*n + 68)*(n - 1)^2*(3*n - 4)^2*(3*n - 5)^2*a(n-2) with a(0) = 1 and a(1) = 13.
a(n) ~ sqrt(17 + 61/sqrt(13)) * ((1921 + 533*sqrt(13))/54)^n / (6*Pi*sqrt(2)*n). - Vaclav Kotesovec, Feb 17 2024
a(n) = Sum_{k = 0..n} binomial(n, k) * binomial(3*n, k) * binomial(3*n+k, k). - Peter Bala, Feb 26 2024

A363869 a(n) = A108625(3n, 2n).

Original entry on oeis.org

1, 55, 12559, 3685123, 1205189519, 418856591055, 151353475289275, 56193989426243199, 21283943385478109071, 8185785098679048061837, 3186604888590691870779559, 1252744279186835597251089055, 496508748101370063304243706939, 198134918989716743103591120933103

Offset: 0

Peter Bala, Jun 27 2023

a(n) = B(3*n, 2*n, 3*n) in the notation of Straub, equation 24. It follows from Straub, Theorem 3.2, that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
For other cases, see A099601 (r = 2, s = 1), A363867 (r = 1, s = 2), A363868 (r = 3, s = 1), A363870 (r = 1, s = 3) and A363871 (r = 2, s = 3).

Paolo Xausa, Table of n, a(n) for n = 0..350
Peter Bala, A recurrence for A363869

Cf. A005258, A099601, A108625, A363864 - A363871.

A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):
seq(simplify(A108625(3*n, 2*n)), n = 0..20);

A363869[n_] := HypergeometricPFQ[{-2*n, -3*n, 3*n + 1}, {1, 1}, 1];
Array[A363869, 20, 0] (* Paolo Xausa, Feb 26 2024 *)

a(n) = Sum_{k = 0..2*n} binomial(3*n, k)^2 * binomial(5*n-k, 3*n).
a(n) = Sum_{k = 0..2*n} (-1)^k * binomial(3*n, k)*binomial(5*n-k, 3*n)^2.
a(n) = hypergeom( [-2*n, -3*n, 3*n+1], [1, 1], 1).
a(n) = [x^(2*n)] 1/(1 - x)*Legendre_P(3*n, (1 + x)/(1 - x)).
a(n) ~ 2^(4*n) * 3^(3*n) / (sqrt(5)*Pi*n). - Vaclav Kotesovec, Apr 27 2024

A363870 a(n) = A108625(n, 3*n).

Original entry on oeis.org

1, 7, 127, 2869, 71631, 1894007, 51978529, 1464209383, 42050906191, 1225778575021, 36156060825127, 1076772406867549, 32324178587781393, 976893529756053501, 29693248490460447747, 907027175886637081619, 27826656707376811715663, 856949305975908664414097

Offset: 0

Peter Bala, Jun 27 2023

a(n) = B(n, 3*n, n) in the notation of Straub, equation 24. It follows from Straub, Theorem 3.2, that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
For other cases, see A099601 (r = 2, s = 1), A363867 (r = 1, s = 2), A363868 (r = 3, s = 1), A363869 (r = 3, s = 2) and A363871 (r = 2, s = 3).

G. C. Greubel, Table of n, a(n) for n = 0..500
Peter Bala, A recurrence for A363870

Cf. A005258, A099601, A108625, A363864 - A363871.

A363870:= func< n | (&+[Binomial(n,j)^2*Binomial(3*n+j,n): j in [0..n]]) >;
[A363870(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):
seq(simplify(A108625(n, 3*n)), n = 0..20);

Table[HypergeometricPFQ[{-n,-3*n,n+1}, {1,1}, 1], {n,0,30}] (* G. C. Greubel, Oct 05 2023 *)

def A363870(n): return sum(binomial(n,j)^2*binomial(3*n+j,n) for j in range(n+1))
[A363870(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

a(n) = Sum_{k = 0..n} binomial(n, k)^2 * binomial(3*n+k, n).
a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n, k)*binomial(3*n+k, n)^2.
a(n) = hypergeometric3F2( [-n, -3*n, n+1], [1, 1], 1).
a(n) = [x^(3*n)] 1/(1 - x)*Legendre_P(n, (1 + x)/(1 - x)).
a(n) ~ sqrt(25 + 151/sqrt(37)) * (11906 + 1961*sqrt(37))^n / (Pi * 2^(3/2) * n * 3^(6*n+1)). - Vaclav Kotesovec, Feb 17 2024
a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k, k)*binomial(3*n, k). - Peter Bala, Feb 25 2024

A363865 a(n) = A143007(3*n,n).

Original entry on oeis.org

1, 25, 2773, 430081, 77620661, 15276834025, 3180268712125, 688612022804773, 153504724110658741, 34994264014955310181, 8120680975872203708773, 1911897036160037674700065, 455553725980571500127902109

Offset: 0

Peter Bala, Jun 25 2023

a(n) = A(3*n,n,3*n,n) in the notation of Straub, equation 8. It follows from Straub, Theorem 1.2, that the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and all positive integers n and k.

More generally, for positive integers r and s the sequence {A143007(r*n, s*n) : n >= 0} satisfies the above supercongruences. For other cases, see A005259 (r = s = 1), A363864 (r = 2, s = 1) and A363866 (r = 3, s = 2).

Peter Bala, A recurrence for A363865
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.

Cf. A005259, A143007, A363864 - A363871.

A143007 := proc(n, k); add(binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j), j = 0..n) end:
seq(A143007(3*n, n), n = 0..20);
# alternative program
seq(simplify(hypergeom([3*n+1, -3*n, n+1, -n], [1, 1, 1], 1)), n = 0..20);

a(n) = Sum_{k = 0..n} binomial(3*n,n-k)^2*binomial(3*n+k,k)^2.
a(n) = Sum_{k = 0..n} binomial(3*n+k,2*k)*binomial(2*k,k)^2*binomial(n+k,2*k).
a(n) = hypergeom([3*n+1, -3*n, n+1, -n], [1, 1, 1], 1)
a(n) = [x^n] 1/(1 - x)*( Legendre_P(3*n,(1 + x)/(1 - x)) )^2 =  [x^(3*n)] 1/(1 - x)*( Legendre_P(n,(1 + x)/(1 - x)) )^2.

A363866 a(n) = A143007(3n,2n).

Original entry on oeis.org

1, 253, 494341, 1403375905, 4684608730309, 17126002734202253, 66366682204430084569, 267832273159817887638881, 1113652383352571992799711941, 4737943697041408831021629805753, 20526206833382185439454748049996341

Offset: 0

Peter Bala, Jun 25 2023

a(n) = A(3*n,2*n,3*n,2*n) in the notation of Straub, equation 8. It follows from Straub, Theorem 1.2, that the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and all positive integers n and k.

More generally, for positive integers r and s, the sequence {A143007(r*n, s*n) : n >= 0} satisfies the above supercongruences. For other cases, see A005259 (r = s = 1), A363864 (r = 2, s = 1) and A363865 (r = 3, s = 1).

Peter Bala, A recurrence for A363866
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.

Cf. A005259, A143007, A363864 - A363871.

A143007 := proc(n, k); add(binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j), j = 0..n) end:
seq(A143007(3*n, 2*n), n = 0..20);
# alternative program
seq(simplify(hypergeom([3*n+1, -3*n, 2*n+1, -2*n], [1, 1, 1], 1)), n = 0..20);

a(n) = Sum_{k = 0..2*n} binomial(3*n+k,2*k)*binomial(2*k,k)^2*binomial(2*n+k,2*k).
a(n) = Sum_{k = 0..2*n} binomial(3*n,k)^2*binomial(5*n-k,2*n-k)^2.
a(n) = hypergeom([3*n+1, -2*n, 3*n+1, -2*n], [1, 1, 1], 1)
a(n) = [x^(2*n)] 1/(1 - x)*( Legendre_P(3*n,(1 + x)/(1 - x)) )^2 =  [x^(3*n)] 1/(1 - x)*( Legendre_P(2*n,(1 + x)/(1 - x)) )^2.

A364244 a(n) = A143007(2*n-1, n-1) for n >= 1.

Original entry on oeis.org

1, 25, 1441, 107353, 9073501, 826861993, 79219824685, 7865844936025, 802198564524325, 83532710607121525, 8844234718023010681, 949244022625120188265, 103044177225432902852641, 11293765432962617876667253, 1248038875078327818254657941

Offset: 1

Peter Bala, Jul 16 2023

The sequence of Apéry numbers A005259 forms the main diagonal of A143007, i.e., A005259(n) = A143007(n, n). The Apéry numbers satisfy the supercongruences A005259(n*p^r) == A005259(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.

More generally, for positive integers r and s, the sequence defined by a(r,s;n) = A143007(r*n - 1, s*n - 1) may also satisfy the same supercongruences. This is the case r = 2, s = 1. Compare with the comments in A363864.

Cf. A005259, A143007, A363864, A364243.

seq( add(binomial(2*n-1, k)^2 * binomial(3*n-2-k, 2*n-1)^2, k = 0..n-1), n = 1..20);
# alternative program
seq(simplify(hypergeom([2*n, 1 - 2*n, n, 1 - n], [1, 1, 1], 1)), n = 1..20);

Table[HypergeometricPFQ[{2*n, 1 - 2*n, n, 1 - n}, {1, 1, 1}, 1], {n, 1, 20}] (* Vaclav Kotesovec, Jul 16 2023 *)

a(n) = Sum_{k = 0..n-1} binomial(2*n-1, k)^2 * binomial(3*n-2-k, 2*n-1)^2.
a(n) = hypergeom([2*n, 1 - 2*n, n, 1 - n], [1, 1, 1], 1).
P-recursive: 2*(n-1)^3*(2*n-1)^3*(440*n^3-2178*n^2+3600*n-1987)*a(n) = (865920*n^9 - 9481824*n^8 + 45492136*n^7 - 125359294*n^6 + 218361816*n^5 - 249018285*n^4 + 185709390*n^3 - 87271191*n^2 + 23447876*n - 2745998)*a(n-1) -  2*(2*n-3)^3*(n-2)^3*(440*n^3-858*n^2+564*n-125)*a(n-2) with a(1) = 1 and a(2) = 25.
a(n) ~ phi^(10*n - 4) / (2^(5/2) * 5^(1/4) * (Pi*n)^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 16 2023

Offset changed by Georg Fischer, Nov 03 2023

cp's OEIS Frontend

A363871 a(n) = A108625(2*n, 3*n).

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A363867 a(n) = A108625(n,2*n).

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A363868 a(n) = A108625(3*n, n).

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A363869 a(n) = A108625(3*n, 2*n).

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A363870 a(n) = A108625(n, 3*n).

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A363865 a(n) = A143007(3*n,n).

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A363866 a(n) = A143007(3*n,2*n).

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A363871 a(n) = A108625(2n, 3n).

A363869 a(n) = A108625(3n, 2n).

A363866 a(n) = A143007(3n,2n).