A339710 -id:A339710 - cp's OEIS frontend

A362722 a(n) = [x^n] ( E(x)/E(-x) )^n where E(x) = exp( Sum_{k >= 1} A005258(k)*x^k/k ).

1, 6, 72, 1266, 23232, 445506, 8740728, 174366114, 3519799296, 71696570010, 1470795168072, 30344633110710, 628994746308288, 13089254107521234, 273292588355096760, 5722454505166750266, 120119862431845048320, 2526922404360157374738, 53260275108329790626952

Offset: 0

Peter Bala, May 01 2023

It is known that the sequence of Apéry numbers A005258 satisfies the Gauss congruences A005258(n*p^r) == A005258(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
One consequence is that the power series expansion of E(x) = exp( Sum_{k
>= 1} A005258(k)*x^k/k ) = 1 + 3*x + 14*x^2 + 82*x^3 + 551*x^4 + ... has integer coefficients (see, for example, Beukers, Proposition, p. 143). Therefore, the power series expansion of E(x)/E(-x) also has integer coefficients and so a(n) = [x^n] ( E(x)/E(-x) )^n is an integer.
In fact, the Apéry numbers satisfy stronger congruences than the Gauss congruences known as supercongruences: A005258(n*p^r) == A005258(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r (see Straub, Section 1).
We conjecture below that {a(n)} satisfies supercongruences similar to (but weaker than) the above supercongruences satisfied by the Apéry numbers.

F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy
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: A005258, A362723 - A362733.

A005258 := proc(n) add(binomial(n, k)^2*binomial(n+k,k), k = 0..n) end proc:
E(n,x) := series(exp(n*add(2*A005258(2*k+1)*x^(2*k+1)/(2*k+1), k = 0..10)), x, 21):
seq(coeftayl(E(n,x), x = 0, n), n = 0..20);

a(n) = [x^n] exp( Sum_{k >= 1} n*( 2*A005258(2*k+1)*x^(2*k+1) )/(2*k+1) ).
Conjectures:
1) the supercongruence a(p^r) == a(p^(r-1)) (mod p^(2*r+1)) holds for all primes p >= 5.
2) for n >= 2, a(n*p) == a(n) (mod p^2) holds for all primes p >= 3.
3) for r >= 2, the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for all primes p >= 3 and n >= 1.

A362723 a(n) = [x^n] ( E(x)/E(-x) )^n where E(x)= exp( Sum_{k >= 1} A005259(k)*x^k/k ).

Original entry on oeis.org

1, 10, 200, 7390, 260800, 10263010, 407520920, 16758685030, 697767370240, 29525605934410, 1261570539980200, 54419751094210270, 2364396136291654720, 103393259758470870770, 4545671563318715532280, 200804420082143353690390, 8907295723280072012247040, 396570344897237949249382010

Offset: 0

Peter Bala, May 01 2023

It is known that the sequence of Apéry numbers A005259 satisfies the Gauss congruences A005259(n*p^r) == A005259(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
One consequence is that the power series expansion of E(x) = exp( Sum_{k >= 1} A005259(k)*x^k/k ) = 1 + 5*x + 49*x^2 + 685*x^3 + 11807*x^4 + ... has integer coefficients. See A267220. For a proof see, for example, Beukers, Proposition, p 143. Therefore, the power series expansion of E(x)/E(-x) also has integer coefficients and so a(n) = [x^n] ( E(x)/E(-x) )^n is an integer.
In fact, the Apéry numbers satisfy stronger congruences than the Gauss congruences known as 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 (see Straub, Section 1).
We conjecture below that {a(n)} satisfies supercongruences similar to (but weaker than) the above supercongruences satisfied by the Apéry numbers.

Frits Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy
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, A267220, A362722 - A362733.

A005259 := proc(n) add(binomial(n, k)^2*binomial(n+k,k)^2, k = 0..n) end;
E(n,x) := series(exp(n*add(2*A005259(2*k+1)*x^(2*k+1)/(2*k+1), k = 0..10)), x, 21):
seq(coeftayl(E(n,x), x = 0, n), n = 0..20);

a(n) = [x^n] exp( Sum_{k >= 1} n*( 2*A005259(2*k+1)*x^(2*k+1) )/(2*k+1) ).
Conjectures:
1) the supercongruence a(p) == a(1) (mod p^3) holds for all primes p >= 5 (checked up to p = 101).
2) for n >= 2, a(n*p) == a(n) (mod p^2) holds for all primes p >= 5.
3) for n >= 1, r >= 2, the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for all primes p >= 5.

A362724 a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A005258(k)*x^k/k ).

Original entry on oeis.org

1, 3, 37, 525, 7925, 123878, 1980199, 32150030, 527984245, 8747075100, 145917510662, 2447835093498, 41253740275559, 697956867712705, 11847510103853090, 201678623730755525, 3441648250114203253, 58859380176953941937, 1008553120517397082420, 17311102730697482426850

Offset: 0

Peter Bala, May 02 2023

Compare with A362722.
It is known that the sequence of Apéry numbers A005258 satisfies the Gauss congruences A005258(n*p^r) == A005258(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
One consequence is that the power series expansion of E(x) = exp( Sum_{k >= 1} A005258(k)*x^k/k ) = 1 + 3*x + 14*x^2 + 82*x^3 + 551*x^4 + ... has integer coefficients (see, for example, Beukers, Proposition, p. 143), and therefore a(n) = [x^n] E(x)^n is an integer.
In fact, the Apéry numbers satisfy stronger congruences than the Gauss congruences known as supercongruences: A005258(n*p^r) == A005258(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r (see Straub, Section 1).
We conjecture below that {a(n)} satisfies supercongruences similar to (but weaker than) the above supercongruences satisfied by the Apéry numbers.
More generally, we inductively define a family of sequences {a(i,n) : n >= 0}, i >= 0, by setting a(0,n) = A005258(n) and, for i >= 1, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. In this notation the present sequence is {a(1,n)}.
We conjecture that the sequences {a(i,n) : n >= 0}, i >= 1, satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(2*r)) for all primes p >= 3, and positive integers n and r.

F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy
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. A005258, A362722 - A362733.

A005258 := proc(n) add(binomial(n,k)^2*binomial(n+k,k), k = 0..n) end proc:
E(n,x) := series(exp(n*add(A005258(k)*x^k/k, k = 1..20)), x, 21):
seq(coeftayl(E(n,x), x = 0, n), n = 0..20);

Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for all primes p >= 3 and positive integers n and r.

A362725 a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A005259(k)*x^k/k ).

Original entry on oeis.org

1, 5, 123, 3650, 118059, 4015380, 141175410, 5082313276, 186243853995, 6920379988871, 260030830600748, 9860709859708350, 376821110248674594, 14494688046084958080, 560708803489098556632, 21797478402692370515400, 851057798310071946207915, 33356751162583463626417872

Offset: 0

Peter Bala, May 02 2023

Compare with A362723.
It is known that the sequence of Apéry numbers A005259 satisfies the Gauss congruences A005259(n*p^r) == A005259(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
One consequence is that the power series expansion of E(x) = exp( Sum_{k
>= 1} A005259(k)*x^k/k ) = 1 + 5*x + 49*x^2 + 685*x^3 + 11807*x^4 + ... has integer coefficients (see, for example, Beukers, Proposition, p. 143), and therefore a(n) = [x^n] E(x)^n is an integer.
In fact, the Apéry numbers satisfy stronger congruences than the Gauss congruences known as 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 (see Straub, Section 1).
We conjecture below that the present sequence satisfies supercongruences similar to (but weaker than) the above supercongruences satisfied by the Apéry numbers.
More generally, we inductively define a family of sequences {a(i,n) : n >= 0}, i >= 0, by setting a(0,n) = A005259(n) and, for i >= 1, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. In this notation the present sequence is {a(1,n)}.
We conjecture that the sequences {a(i,n) : n >= 0}, i >= 1, satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(2*r)) for all primes p >= 3, and positive integers n and r.

F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy
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, A362722 - A362733.

A005259 := proc(n) add(binomial(n,k)^2*binomial(n+k,k)^2, k = 0..n) end proc:
E(n,x) := series(exp(n*add((A005259(k)*x^k)/k, k = 1..20)), x, 21):
seq(coeftayl(E(n,x), x = 0, n), n = 0..20);

Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for all primes p >= 3 and positive integers n and r.

A362731 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A000172(k)*x^k/k ).

Original entry on oeis.org

1, 2, 18, 182, 1954, 21702, 246366, 2839846, 33105186, 389264798, 4608481918, 54862022910, 656099844526, 7876525155020, 94867757934870, 1145843922848232, 13873839714404642, 168345900709550388, 2046612356962697502, 24923311881995950740, 303974276349311203854

Offset: 0

Peter Bala, May 05 2023

It is known that the sequence of Franel numbers A000172 satisfies the Gauss congruences A000172(n*p^r) == A000172(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
One consequence is that the power series expansion of E(x) = exp( Sum_{k >= 1} A000172(k)*x^k/k ) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + ... (the g.f. of A166990) has integer coefficients (see, for example, Beukers, Proposition, p. 143). Therefore a(n) = [x^n] E(x)^n is an integer.
In fact, the Franel numbers satisfy stronger congruences than the Gauss congruences known as supercongruences: A000172(n*p^r) == A000172(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r.

F. Beukers, Some congruences for the Apery numbers, Journal of Number Theory, Vol. 21, Issue 2, Oct. 1985, pp. 141-155. local copy

Cf. A000172, A166990, A362722 - A362733.

A000172 := proc(n) add(binomial(n,k)^3, k = 0..n); end:
E(n,x) := series( exp(n*add(A000172(k)*x^k/k, k = 1..20)), x, 21 ):
seq(coeftayl(E(n,x), x = 0, n), n = 0..20);

The Gauss congruence a(n*p^r) == a(n*p^(r-1)) (mod p^r) holds for all primes p and positive integers n and r.
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for
all primes p and positive integers n and r.

A370258 Triangle read by rows: T(n, k) = binomial(n, k)binomial(2n+k, k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 3, 1, 10, 15, 1, 21, 84, 84, 1, 36, 270, 660, 495, 1, 55, 660, 2860, 5005, 3003, 1, 78, 1365, 9100, 27300, 37128, 18564, 1, 105, 2520, 23800, 107100, 244188, 271320, 116280, 1, 136, 4284, 54264, 339150, 1139544, 2089164, 1961256, 735471, 1, 171, 6840, 111720, 921690, 4239774, 11306064

Offset: 0

Peter Bala, Feb 13 2024

Compare with A063007(n, k) = binomial(n, k)*binomial(n+k, k), the table of coefficients of the shifted Legendre polynomials P(n, 2*x + 1).

			Triangle begins
n\k| 0    1     2      3       4       5       6       7
- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 0 | 1
 1 | 1    3
 2 | 1   10    15
 3 | 1   21    84     84
 4 | 1   36   270    660     495
 5 | 1   55   660   2860    5005    3003
 6 | 1   78  1365   9100   27300   37128   18564
 7 | 1  105  2520  23800  107100  244188  271320  116280
 ...

A114496 (row sums), A000984 (alt. row sums unsigned), A005809 (main diagonal), A090763 (first subdiagonal), A014105 (column 1).

Cf. A026000, A063007, A102537, A110608, A339710.

seq(print(seq(binomial(n, k)*binomial(2*n+k, k), k = 0..n)), n = 0..10);

n-th row polynomial R(n, x) = Sum_{k = 0..n} binomial(n, k)*binomial(2*n+k, k)*x^k = (1 + x)^n * Sum_{k = 0..n} binomial(n, k)*binomial(2*n, k)*(x/(1 + x))^k = Sum_{k = 0..n} A110608(n, n-k)*x^k*(1 + x)^(n-k).
(x - 1)^n * R(n, 1/(x - 1)) = Sum_{k = 0..n} binomial(n,k)*binomial(2*n, n-k)*x^k = the n-th row polynomial of A110608.
R(n, x) = hypergeom([-n, 2*n + 1], [1], -x).
Second-order differential equation: ( (1 + x)^n * (x + x^2)*R(n, x)' )' = n*(2*n + 1)*(1 + x)^n * R(n, x), where the prime indicates differentiation w.r.t. x.
Equivalently, x*(1 + x)*R(n, x)'' + ((n + 2)*x + 1)*R(n, x)' - n*(2*n + 1)*R(n, x)' = 0.
Analog of Rodrigues' formula for the shifted Legendre polynomials:
R(n, x) = 1/(1 + x)^n * 1/n! * (d/dx)^n (x*(1 + x)^2)^n.
Analog of Rodrigues' formula for the Legendre polynomials:
R(n, (x-1)/2) = 1/(n!*2^n) * 1/(1 + x)^n *(d/dx)^n ((x - 1)*(x + 1)^2)^n.
Orthogonality properties:
Integral_{x = -1..0} (1 + x)^n * R(n, x) * R(m, x) dx = 0 for n > m.
Integral_{x = -1..0} (1 + x)^n * R(n, x)^2 dx = 1/(3*n + 1).
Integral_{x = -1..0} (1 + x)^(n+m) * R(n, x) * R(m, x) dx = 0 for m >= 2*n + 1 or m <= (n - 1)/2.
Integral_{x = -1..0} (1 + x)^k * R(n, x) dx = 0 for n <= k <= 2*n - 1;
Integral_{x = -1..0} (1 + x)^(2*n) * R(n, x) dx = (2*n)!*n!/(3*n+1)! = 1/A090816(n).
Recurrence for row polynomials:
2*n*(2*n - 1)*((9*n - 12)*x + 8*n - 11)*(1 + x)*R(n, x) = (9*(3*n - 1)*(3*n - 2)*(3*n - 4)*x^3 + 3*(3*n - 1)*(3*n - 2)*(20*n - 27)*x^2 + 6*(3*n - 2)*(20*n^2 - 34*n + 9)*x + 2*(32*n^3 - 76*n^2 + 50*n - 9))*R(n-1, x) - 2*(n - 1)*(2*n - 3)*((9*n - 3)*x + 8*n - 3)*R(n-2, x), with R(0, x) = 1, R(1, x) = 1 + 3*x.
Conjecture: exp( Sum_{n >= 1} R(n,t)*x^n/n ) = 1 + (1 + 3*t)*x + (1 + 8*t + 12*t^2)*x^2 + ... is the o.g.f. for A102537. If true, then it would follows that, for each integer t, the sequence u = {R(n,t) : n >= 0} satisfies the Gauss congruences u(m*p^r) == u(m*p^(r-1)) (mod p^r) for all primes p and positive integers m and r.
R(n, 1) = A114496(n); R(n, -1) = (-1)^n * A000984(n).
R(n, 2) = A339710(n); R(n, -2) = (-1)^n * A026000(n).
(2^n)*R(n, -1/2) = A234839(n).

cp's OEIS Frontend

A362722 a(n) = [x^n] ( E(x)/E(-x) )^n where E(x) = exp( Sum_{k >= 1} A005258(k)*x^k/k ).

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A362723 a(n) = [x^n] ( E(x)/E(-x) )^n where E(x)= exp( Sum_{k >= 1} A005259(k)*x^k/k ).

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A362724 a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A005258(k)*x^k/k ).

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A362725 a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A005259(k)*x^k/k ).

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A362731 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A000172(k)*x^k/k ).

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A370258 Triangle read by rows: T(n, k) = binomial(n, k)*binomial(2*n+k, k), 0 <= k <= n.

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A370258 Triangle read by rows: T(n, k) = binomial(n, k)binomial(2n+k, k), 0 <= k <= n.