A274783 -id:A274783 - cp's OEIS frontend

A361637 Constant term in the expansion of (1 + x + y + z + 1/(xyz))^n.

1, 1, 1, 1, 25, 121, 361, 841, 4201, 25705, 118441, 423721, 1628881, 8065201, 41225185, 184416961, 768211081, 3420474121, 16620237001, 79922011465, 364149052705, 1638806098945, 7655390077105, 36739991161105, 174363209490625, 811840219629121, 3790118889635521

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Diagonal of the rational function 1/(1 - (x^4 + y^4 + z^4 + w^4 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Winston de Greef, Table of n, a(n) for n = 0..1429

Cf. A002426, A344560, A361703.
Cf. A008977, A328725.
Cf. A082488, A274783, A274785.

a(n) = n!*sum(k=0, n\4, 1/(k!^4*(n-4*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} 1/(k!^4 * (n-4*k)!).
G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(4*k)/(1-x)^(4*k+1).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) - (n-1)*(6*n^2 - 12*n + 7)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 255*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 5^(n + 3/2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). (End)

A208425 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-x)^(3n+1).

Original entry on oeis.org

1, 1, 7, 25, 151, 751, 4411, 24697, 146455, 862351, 5195257, 31392967, 191815339, 1177508515, 7276161907, 45154764025, 281492498455, 1761076827895, 11055132835705, 69600761349175, 439370198255401, 2780265190892641, 17631718101804517, 112038660509078695

Offset: 0

Paul D. Hanna, Feb 26 2012

nonn

Compare g.f. to: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1), which is a g.f. of the Franel numbers (A000172).
From Zhi-Wei Sun, Nov 12 2016: (Start)
Conjecture: (i) For any prime p > 3 and positive integer n, the number (a(p*n)-a(n))/(p*n)^3 is always a p-adic integer.
(ii) For any prime p == 1 (mod 3), we have Sum_{k=0..p-1}a(k) == C(2(p-1)/3,(p-1)/3) (mod p^2). For any prime p == 2 (mod 3), we have Sum_{k=0..p-1}a(k) == 2p/C(2(p+1)/3,(p+1)/3) (mod p^2).
We have proved part (i) of this conjecture for n = 1. (End)
Diagonal of rational functions 1/(1 - x*y - y*z - x*z - x*y*z), 1/(1 - x*y + y*z + x*z - x*y*z). - Gheorghe Coserea, Jul 03 2018
Number of paths from (0,0,0) to (n,n,n) using steps (1,1,0), (1,0,1), (0,1,1), and (1,1,1). - William J. Wang, Dec 07 2020
Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + x*y*z)). - Seiichi Manyama, Jul 04 2025

			G.f.: A(x) = 1 + x + 7*x^2 + 25*x^3 + 151*x^4 + 751*x^5 + 4411*x^6 +...
where
A(x) = 1/(1-x) + 6*x^2/(1-x)^4 + 90*x^4/(1-x)^7 + 1680*x^6/(1-x)^10 + 34650*x^8/(1-x)^13 + 756756*x^10/(1-x)^16 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
A. Bostan, S. Boukraa, J.-M. Maillard and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Hao Pan and Zhi-Wei Sun, Supercongruences for central trinomial coefficients, arXiv:2012.05121 [math.NT], 2020.
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A000172, A001850, A208426, A244973, A274783.

Cf. A081798, A344560.

series(hypergeom([1/3, 2/3], [1], 27*x^2/(1 - x)^3)/(1 - x), x=0, 25): seq(coeff(%, x, n), n=0..23);  # Mark van Hoeij, May 20 2013
a := n -> hypergeom([1/2 - n/2, -n/2, n + 1], [1, 1], 4); seq(simplify(a(n)), n=0..23);  # Peter Luschny, Jan 11 2025

nmax = 20; CoefficientList[Series[Sum[(3*n)!/n!^3 * x^(2*n)/(1-x)^(3*n+1), {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2016 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)/(1-x+x*O(x^n))^(3*m+1)),n)}
for(n=0,25,print1(a(n),", "))

Conjecture: n^2*(3*n-5)*a(n) +(-9*n^3+24*n^2-17*n+4) *a(n-1) -(3*n-4) *(24*n^2-56*n+27)*a(n-2) -(3*n-2)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Mar 10 2016
a(n) ~ sqrt(1/2 + sqrt(13)*cos(arctan(53*sqrt(3)/19)/3)/6) * (1 + 6*cos(Pi/9))^n / (Pi*n). - Vaclav Kotesovec, Jul 05 2016
It is easy to show that a(n) = Sum_{k=0..n}C(n,k)*C(n-k,k)*C(n+k,k) = Sum_{k=0..n}C(n+k,k)*C(n,2k)*C(2k,k). By this formula and the Zeilberger algorithm, we confirm the recurrence conjectured by R. J. Mathar. - Zhi-Wei Sun, Nov 12 2016
G.f. y=A(x) satisfies: 0 = x*(x + 2)*(x^3 + 24*x^2 + 3*x - 1)*y'' + (3*x^4 + 56*x^3 + 147*x^2 + 12*x - 2)*y' + (x^3 + 9*x^2 + 42*x + 2)*y. - Gheorghe Coserea, Jul 03 2018
a(n) = hypergeom([1/2 - n/2, -n/2, n + 1], [1, 1], 4). - Peter Luschny, Jan 11 2025

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

Original entry on oeis.org

1, 1, 1, 1, 121, 721, 2521, 6721, 128521, 1277641, 7539841, 32527441, 281835841, 3031468441, 23779315561, 139431015361, 962322302761, 9034098300361, 79726215362761, 569831799431881, 3952559737085401, 32660742079719601, 289694072383115401

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Diagonal of the rational function 1/(1 - (v^4 + w^4 + x^4 + y^4 + z^4 + v*w*x*y*z)). - Seiichi Manyama, Jul 04 2025

Winston de Greef, Table of n, a(n) for n = 0..1064

Cf. A001850, A208425, A274783.
Cf. A008978.
Cf. A082489, A361703.

Table[Sum[(n + k)!/(k!^5*(n - 4*k)!), {k, 0, n/4}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 19 2023 *)

a(n) = sum(k=0, n\4, (n+k)!/(k!^5*(n-4*k)!));

a(n) = Sum_{k=0..floor(n/4)} (n+k)!/(k!^5 * (n-4*k)!).
G.f.: Sum_{k>=0} (5*k)!/k!^5 * x^(4*k)/(1-x)^(5*k+1).
Recurrence: n^4*(5*n - 19)*(5*n - 18)*(5*n - 17)*(5*n - 14)*(5*n - 13)*(5*n - 9)*a(n) = (5*n - 19)*(5*n - 18)*(5*n - 14)*(625*n^7 - 6125*n^6 + 23025*n^5 - 43195*n^4 + 45394*n^3 - 28716*n^2 + 10144*n - 1536)*a(n-1) - (5*n - 19)*(31250*n^9 - 568750*n^8 + 4441875*n^7 - 19516000*n^6 + 53172025*n^5 - 93366740*n^4 + 106140132*n^3 - 75781664*n^2 + 30987264*n - 5529600)*a(n-2) + (5*n - 4)*(31250*n^9 - 725000*n^8 + 7354375*n^7 - 42784750*n^6 + 157237100*n^5 - 378480620*n^4 + 596812963*n^3 - 594970390*n^2 + 340845072*n - 85743360)*a(n-3) + (5*n - 16)*(5*n - 9)*(5*n - 8)*(5*n - 4)*(78000*n^6 - 1450800*n^5 + 11179085*n^4 - 45672814*n^3 + 104341702*n^2 - 126378083*n + 63400710)*a(n-4) + (n-4)^4*(5*n - 14)*(5*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 8)*(5*n - 4)*a(n-5). - Vaclav Kotesovec, Mar 19 2023

A274785 Diagonal of the rational function 1/(1-(wxy*z + wxz + wy + xy + z)).

Original entry on oeis.org

1, 1, 25, 121, 2881, 23521, 484681, 5223625, 97949041, 1243490161, 22061635465, 309799010665, 5331441539425, 79799232449665, 1352284119871465, 21095036702450281, 355125946871044561, 5694209222592780625, 95705961654403180201, 1563714140278617173641, 26311422169994777663761

Offset: 0

Gheorghe Coserea, Jul 13 2016

Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + w^2 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Seiichi Manyama, Table of n, a(n) for n = 0..801
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
S. Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A268545-A268555.

Cf. A082488, A274783, A361637.

seq(add(binomial(n+2*k, 2*k)*binomial(n, 2*k)*binomial(2*k, k)^2, k = 0..floor(n/2)), n = 0..20); # Peter Bala, Jan 27 2018

Table[Sum[Binomial[n + 2*k, 2*k]*Binomial[n, 2*k]*Binomial[2*k, k]^2, {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*y*z+w*x*z+w*y+x*y+z));
diag(n, expr, var) = {
  my(a = vector(n));
  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
  for (k = 1, n, a[k] = expr;
       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
  return(a);
};
diag(12, R, [x,y,z,w])

a(n) = sum(k=0, n\2, binomial(n + 2*k,2*k) * binomial(n,2*k) * binomial(2*k,k)^2) \\ Andrew Howroyd, Mar 18 2023

0 = (-x^2+2*x^3+257*x^4+508*x^5+257*x^6+2*x^7-x^8)*y''' + (-3*x+15*x^2+1524*x^3+2286*x^4+789*x^5+3*x^6-6*x^7)*y'' + (-1+16*x+1687*x^2+1168*x^3+217*x^4-8*x^5-7*x^6)*y' + (1+183*x-178*x^2-2*x^3-3*x^4-x^5)*y, where y is the g.f.
a(n) = Sum_{k = 0..floor(n/2)} C(n + 2*k,2*k)*C(n,2*k)*C(2*k,k)^2 (apply Eger, Theorem 3 to the set of column vectors S = {[0,0,1,0], [1,1,0,0], [0,1,0,1], [1,0,1,1],[1,1,1,1]}). - Peter Bala, Jan 27 2018
n^3*(n - 2)*(2*n - 5)*a(n) = (2*n - 5)*(2*n - 1)*(2*n^3 - 6*n^2 + 4*n - 1)*a(n-1) + (2*n - 3)*(250*n^4 - 1500*n^3 + 3066*n^2 - 2448*n + 629)*a(n-2) + (2*n - 5)*(2*n - 1)*(2*n^3 - 12*n^2 + 22*n - 11)*a(n-3) - (2*n - 1)*(n - 1)*(n - 3)^3*a(n-4). - Peter Bala, Mar 17 2023
a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * Pi^(3/2) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 17 2023

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A361637 Constant term in the expansion of (1 + x + y + z + 1/(x*y*z))^n.

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A208425 Expansion of Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-x)^(3*n+1).

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A361636 Diagonal of the rational function 1/(1 - v*w*x*y*z * (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

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A274785 Diagonal of the rational function 1/(1-(w*x*y*z + w*x*z + w*y + x*y + z)).

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A361637 Constant term in the expansion of (1 + x + y + z + 1/(xyz))^n.

A208425 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-x)^(3n+1).

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

A274785 Diagonal of the rational function 1/(1-(wxy*z + wxz + wy + xy + z)).