cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Mar 19 2023

Keywords

Comments

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

Crossrefs

Programs

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

Formula

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)

A274783 Diagonal of the rational function 1/(1 - (w*x*y*z + w*x*y + w*x*z + w*y*z + x*y*z)).

Original entry on oeis.org

1, 1, 1, 25, 121, 361, 3361, 24361, 116425, 790441, 6060121, 36888721, 238815721, 1760983225, 11968188961, 79763351305, 570661612585, 4040282139625, 27901708614985, 198090585115105, 1420583920034161, 10056659775872161, 71730482491962361, 517012699162717825, 3713833648541268121
Offset: 0

Views

Author

Gheorghe Coserea, Jul 13 2016

Keywords

Comments

Diagonal of the rational function 1/(1 - (x^3 + y^3 + z^3 + w^3 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Crossrefs

Programs

  • Maple
    with(combinat):
    seq(add((n+k)!/(k!^4*(n-3*k)!), k = 0..floor(n/3)), n = 0..20); # Peter Bala, Jan 27 2018
  • PARI
    my(x='x, y='y, z='z, w='w);
    R = 1/(1-(w*x*y*z+w*x*y+w*x*z+w*y*z+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(20, R, [x,y,z,w])

Formula

0 = x^2*(x+3)^2*(x^4 - 260*x^3 + 6*x^2 - 4*x + 1)*y''' + 3*x*(x+3)*(2*x^5 - 381*x^4 - 1944*x^3 + 34*x^2 - 18*x + 3)*y'' + (7*x^6 - 764*x^5 - 9101*x^4 - 27264*x^3 + 381*x^2 - 132*x + 9)*y' + (x^5 - 13*x^4 - 246*x^3 - 5946*x^2 + 69*x - 9)*y, where y is the g.f.
a(n) = Sum_{k = 0..floor(n/3)} (n+k)!/(k!^4*(n-3*k)!) = Sum_{k = 0..floor(n/3)} binomial(n,3*k)*binomial(n+k,k)*(3*k)!/k!^3 (apply Eger, Theorem 3 to the set of column vectors S = {[1,1,1,1], [1,1,0,1], [1,0,1,1], [0,1,1,1], [1,1,1,0]}). - Peter Bala, Jan 27 2018
G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(3*k)/(1-x)^(4*k+1). - Seiichi Manyama, Mar 19 2023
From Vaclav Kotesovec, Mar 19 2023: (Start)
Recurrence: n^3*(2*n - 5)*(4*n - 11)*(4*n - 7)*a(n) = (4*n - 11)*(32*n^5 - 184*n^4 + 368*n^3 - 327*n^2 + 147*n - 27)*a(n-1) - (192*n^6 - 1920*n^5 + 7628*n^4 - 15366*n^3 + 16567*n^2 - 9117*n + 2025)*a(n-2) + (4*n - 9)*(4*n - 3)*(520*n^4 - 4420*n^3 + 13809*n^2 - 18769*n + 9367)*a(n-3) - (n-3)^3*(2*n - 3)*(4*n - 7)*(4*n - 3)*a(n-4).
a(n) ~ sqrt(9/8 + 3/(32*sqrt(2)) + sqrt(1085/32 + 161/(2*sqrt(2)))/8) * (1 + 2*sqrt(2) + 2*sqrt(2*(2*sqrt(2) - 1)))^n / (Pi^(3/2) * n^(3/2)). (End)
Showing 1-2 of 2 results.