A351858 -id:A351858 - cp's OEIS frontend

A372211 a(n) = [x^n] f(x)^n, where f(x) = (1 - x^5)^5/((1 - x^2)^2 * (1 - x^3)^3).

1, 0, 4, 9, 36, 125, 535, 1715, 7716, 26739, 111379, 419265, 1683351, 6518499, 26081381, 102089384, 408200740, 1612289384, 6441151477, 25602561864, 102352339411, 408402686750, 1635036583239, 6541552959219, 26227281703575, 105151396500125, 422159487904405, 1695369986497917

Offset: 0

Peter Bala, Apr 22 2024

Let G(x) be a formal power series with integer coefficients. The sequence defined by g(n) = [x^n] G(x)^n satisfies the Gauss congruences: g(n*p^r) == g(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
We conjecture that in this case the stronger supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. Some examples are given below. Cf. A351858.
More generally, if r is a positive integer and s an integer then the sequence defined by u(r,s; n) = [x^(r*n)] f(x)^(s*n) may satisfy the same supercongruences for primes p >= 7.
Even more generally, if a, b and m are a positive integers, with m = a + b, then the sequences whose n-th term equals [x^n] (1 - x^m)^m/((1 - x^a)^a * (1 - x^b)^b) or [x^n] (1 - x^m)^m/((1 + x^a)^a * (1 + x^b)^b) may both satisfy the above supercongruences for sufficiently large primes p depending on m.
The sequence of central binomial coefficients A000984 corresponds to the case m = 2 and a = b = 1.

			Supercongruences:
a(11) = 419265 = (3^2)*5*7*11^3 == 0 (mod 11^3).
a(23) = 6541552959219 = (3^2)*(23^3)*59738573 == 0 (mod 23^3).
a(2*7) - a(2) = 26081381 - 4 = (7^3)*76039 == 0 (mod 7^3).

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Cf. A000984, A351858, A372212, A372213.

f(x) := (1 - x^5)^5/((1 - x^2)^2*(1 - x^3)^3):
seq(coeftayl(f(x)^n, x = 0, n), n = 0..27);

The o.g.f. A(x) = 1 + 4*x^2 + 9*x^3 + 36*x^4 + ... is the diagonal of the bivariate rational function 1/(1 - t*f(x)) and hence is an algebraic function over the field of rational functions Q(x) by Stanley, Theorem 6.33, p. 197.

A372212 a(n) = [x^n] f(x)^n, where f(x) = (1 - x^7)^7/((1 - x^2)^2 * (1 - x^5)^5).

Original entry on oeis.org

1, 0, 4, 0, 36, 25, 364, 441, 3876, 6561, 43779, 91839, 513900, 1245699, 6201199, 16645750, 76379940, 220760742, 955328863, 2916666288, 12090544611, 38466060066, 154437142545, 506976137710, 1987270052460, 6681958793775, 25724578443321, 88104794553729

Offset: 0

Peter Bala, Apr 22 2024

Let G(x) be a formal power series with integer coefficients. The sequence defined by g(n) = [x^n] G(x)^n satisfies the Gauss congruences: g(n*p^r) == g(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
We conjecture that in this case the stronger supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for primes p >= 11 and positive integers n and r. Some examples are given below. Cf. A351858.
More generally, if r is a positive integer and s an integer then the sequence defined by  u(r,s; n) = [x^(r*n)] f(x)^(s*n) may satisfy the same supercongruences.

			Supercongruences:
a(11) = 91839 = 3*(11^3)*23 == 0 (mod 11^3).
a(2*11) - a(2) = 154437142545 - 4 = (11^3)*2671*43441 == 0 (mod 11^3).

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Cf. A351858, A372211, A372213.

f(x) := (1 - x^7)^7/((1 - x^2)^2*(1 - x^5)^5):
seq(coeftayl(f(x)^n, x = 0, n), n = 0..27);

The o.g.f. A(x) = 1 + 4*x^2 + 36*x^4 + 25*x^5 + ... is the diagonal of the bivariate rational function 1/(1 - t*f(x)) and hence is an algebraic function over the field of rational functions Q(x) by Stanley, Theorem 6.33, p. 197.

A372213 a(n) = [x^n] f(x)^n, where f(x) = (1 - x^7)^7/((1 - x^3)^3 * (1 - x^4)^4).

Original entry on oeis.org

1, 0, 0, 9, 16, 0, 171, 539, 528, 3654, 16500, 29282, 101851, 483340, 1215445, 3416634, 14564880, 44585475, 124007202, 462804166, 1555048516, 4547401595, 15500748802, 53459717443, 164998563675, 538593687500, 1845162146828, 5920282930815, 19091999953749, 64389113743812, 211137579083046

Offset: 0

Peter Bala, Apr 22 2024

Let G(x) be a formal power series with integer coefficients. The sequence defined by g(n) = [x^n] G(x)^n satisfies the Gauss congruences: g(n*p^r) == g(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
We conjecture that in this case the stronger supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for primes p >= 11 and positive integers n and r. Some examples are given below. Cf. A351858.
More generally, if r is a positive integer and s an integer then the sequence defined by  u(r,s; n) = [x^(r*n)] f(x)^(s*n) may satisfy the same supercongruences.

			Supercongruences:
a(11) = 29282 = 2*(11^4) == 0 (mod 11^4).
a(13) = 483340 = (2^2)*5*11*(13^3) == 0 (mod 13^3).
a(2*11) = 15500748802 = 2*7*(11^4)*47*1609 == 0 (mod 11^4).

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Cf. A351858, A372211, A372212.

f(x) := (1 - x^7)^7/((1 - x^3)^3*(1 - x^4)^4):
seq(coeftayl(f(x)^n, x = 0, n), n = 0..30);

The o.g.f. A(x) = 1 + 9*x^3 + 16*x^4 + 171*x^6 + ... is the diagonal of the bivariate rational function 1/(1 - t*f(x)) and hence is an algebraic function over the field of rational functions Q(x) by Stanley, Theorem 6.33, p. 197.

a(28) corrected by and more terms from Georg Fischer, Jul 28 2025

A351859 a(n) = [x^n] (1 + x + x^2 + x^3)^(4n)/(1 + x + x^2)^(3n).

Original entry on oeis.org

1, 1, 3, 19, 67, 251, 1137, 4803, 20035, 87013, 377753, 1634469, 7134385, 31261114, 137121113, 603206144, 2660097603, 11749336328, 51981371895, 230336544210, 1021976441817, 4539784391763, 20188837618799, 89871081815631, 400427435522737, 1785639575031501

Offset: 0

Peter Bala, Mar 01 2022

This sequence is the third in an infinite family of sequences defined as follows. Let k be a positive integer. Define the rational function G_k(x) = (1 + x + ... + x^k)^(k+1)/(1 + x + ... + x^(k-1))^k, so that G_1(x) = (1 + x)^2, and define the sequence u_k by u_k(n) = [x^n] G_k(x)^n. See A000984, the sequence of central binomial coefficients, for the case k = 1 and A351858 for the case k = 2. The present sequence is the case k = 3.
Given a power series G(x) with integer coefficients it is known that the sequence (g(n))n>=1 defined by g(n) := [x^n] G(x)^n satisfies the Gauss congruences g(n*p^r) == g(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Thus a(n) satisfies the Gauss congruences. Calculation suggests that, in fact, the stronger supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for primes p >= 5 and positive integers n and r. These supercongruences are known to hold for the central binomial coefficients A000984(n) = [x^n] ((1 + x)^2)^n (Meštrović, equation 39).
More generally, if r is a positive integer and s an integer then the sequence defined by a(r,s;n) = [x^(r*n)] G_3(x)^(s*n) may satisfy the same supercongruences.

			Examples of supercongruences:
a(5) - a(1) = 251 - 1 = 2*(5^3) == 0 (mod 5^3)
a(2*7) - a(2) = 137121113 - 3 = 2*5*(7^4)*5711 == 0 (mod 7^4)
a(5^2) - a(5) = 1785639575031501 - 251 = 2*(5^6)*1373*3989*10433 == 0 (mod 5^6)

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Paolo Xausa, Table of n, a(n) for n = 0..425
R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2011.

Cf. A000984, A103885, A333090, A333564, A333565, A333579, A333715, A351856, A351857, A351858.

seq(add(add(add((-1)^j*binomial(4*n,n-2*i-j-k)*binomial(4*n,i)* binomial(3*n+j-1,j)*binomial(j,k), k = 0..j), j = 0..n), i = 0..n), n = 0..25);

A351859[n_] := Sum[(-1)^j*Binomial[4*n, n-2*i-j-k]*Binomial[4*n, i]*Binomial[3*n+j-1, j]*Binomial[j, k], {i, 0, n}, {j, 0, n}, {k, 0, j}];
Array[A351859, 25, 0] (* Paolo Xausa, May 30 2025 *)

a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,j,(-1)^j*binomial(4*n,n-2*i-j-k)*binomial(4*n,i)*binomial(3*n+j-1,j)*binomial(j,k))));
vector(25,n,a(n-1)) \\ Paolo Xausa, May 04 2022

a(n) = Sum_{i = 0..n} Sum_{j = 0..n} Sum_{k = 0..j} (-1)^j* C(4n,n-2*i-j-k) *C(4n,i)*C(3n+j-1,j)*C(j,k).
The o.g.f. A(x) = 1 + x + 3*x^2 + 19*x^3 + 67*x^4 + ... is the diagonal of the bivariate rational function 1/(1 - t*(1 + x + x^2 + x^3)^4/(1 + x + x^2)^3) and hence is an algebraic function over the field of rational functions Q(x) by Stanley, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*(1 + x + x^2)^3/(1 + x + x^2 + x^3)^4 ) = 1 + x + 2*x^2 + 8*x^3 + 25*x^4 + 81*x^5 + 305*x^6 + .... Then A(x) = 1 + x*F'(x)/F(x).

A372382 Coefficient of x^n in the expansion of ( (1+x+x^2)^4 / (1+x)^3 )^n.

Original entry on oeis.org

1, 1, 9, 25, 169, 651, 3801, 17053, 93225, 450844, 2396859, 12043494, 63354649, 324888305, 1704137493, 8839907475, 46383701545, 242285478474, 1273274074020, 6681277302239, 35178613785819, 185187072845569, 976888169385302, 5154978257816280, 27240094648199961

Offset: 0

Seiichi Manyama, Apr 29 2024

nonn

Cf. A002426, A351858, A372370.

Cf. A372383.

a[n_]:=SeriesCoefficient[((1+x+x^2)^4/(1+x)^3)^n,{x,0,n}]; Array[a,25,0] (* Stefano Spezia, Apr 30 2024 *)

a(n, s=2, t=4, u=-3) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1+x)^3 / (1+x+x^2)^4 ). See A372383.

A372370 Coefficient of x^n in the expansion of ( (1+x+x^2)^2 / (1+x) )^n.

Original entry on oeis.org

1, 1, 5, 13, 53, 176, 677, 2451, 9333, 34978, 133580, 508806, 1953701, 7509178, 28981643, 112046213, 434289525, 1686080622, 6557830310, 25542229740, 99622788428, 389023326600, 1520817551742, 5951305115982, 23310374278437, 91380414955176, 358506409488102

Offset: 0

Seiichi Manyama, Apr 28 2024

nonn

Cf. A027908, A370159, A370160.
Cf. A002426, A351858, A372382.
Cf. A166135.

a[n_]:=SeriesCoefficient[((1+x+x^2)^2/(1+x))^n,{x,0,n}]; Array[a,27,0] (* Stefano Spezia, Apr 30 2024 *)

a(n, s=2, t=2, u=-1) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1+x) / (1+x+x^2)^2 ).

A372369 Coefficient of x^n in the expansion of ( (1+x+x^2)^3 / (1+x) )^n.

Original entry on oeis.org

1, 2, 12, 65, 388, 2352, 14565, 91289, 577764, 3683459, 23621462, 152203482, 984598741, 6390596591, 41596873869, 271424778015, 1774892605284, 11628321367815, 76311803660025, 501554760288813, 3300889231760238, 21750690436059188, 143481522241226962

Offset: 0

Seiichi Manyama, Apr 28 2024

nonn

Cf. A351858, A370170, A370171, A370172.

Cf. A347953.

a(n, s=2, t=3, u=-1) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n,k) * binomial(2*n-k,n-2*k).

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1+x) / (1+x+x^2)^3 ).

cp's OEIS Frontend

A372211 a(n) = [x^n] f(x)^n, where f(x) = (1 - x^5)^5/((1 - x^2)^2 * (1 - x^3)^3).

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A372212 a(n) = [x^n] f(x)^n, where f(x) = (1 - x^7)^7/((1 - x^2)^2 * (1 - x^5)^5).

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A372213 a(n) = [x^n] f(x)^n, where f(x) = (1 - x^7)^7/((1 - x^3)^3 * (1 - x^4)^4).

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A351859 a(n) = [x^n] (1 + x + x^2 + x^3)^(4*n)/(1 + x + x^2)^(3*n).

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A372382 Coefficient of x^n in the expansion of ( (1+x+x^2)^4 / (1+x)^3 )^n.

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A372370 Coefficient of x^n in the expansion of ( (1+x+x^2)^2 / (1+x) )^n.

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A372369 Coefficient of x^n in the expansion of ( (1+x+x^2)^3 / (1+x) )^n.

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A351859 a(n) = [x^n] (1 + x + x^2 + x^3)^(4n)/(1 + x + x^2)^(3n).