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.

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A211799 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 4, 5, 1, 0, 10, 13, 5, 1, 0, 20, 26, 14, 5, 1, 0, 35, 48, 29, 14, 5, 1, 0, 56, 78, 53, 30, 14, 5, 1, 0, 84, 119, 88, 55, 30, 14, 5, 1, 0, 120, 173, 134, 90, 55, 30, 14, 5, 1, 0, 165, 240, 195, 138, 91, 55, 30, 14, 5, 1, 0, 220, 323, 270, 201, 139, 91
Offset: 1

Views

Author

Clark Kimberling, Apr 21 2012

Keywords

Comments

Row 1: A002292
Row 2: A211637
Row 3: A211651
Limiting row sequence: A000330
Let R be the array in A211796 and let R' be the array in A211799. Then R(k,n)+R'(k,n)=3^(n-1).
See the Comments at A211790.

Examples

			Northwest corner:
0...0...1...4....10...20...35...56
0...1...5...13...26...48...78...119
0...1...5...14...29...53...88...134
0...1...5...14...30...55...90...138
0...1...5...14...30...55...91...139

Crossrefs

Cf. A211790.

Programs

  • Mathematica
    z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[w^k > x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A000292 *)
Table[t[2, n], {n, 1, z}]  (* A211637 *)
Table[t[3, n], {n, 1, z}]  (* A211651 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1],
    {n, 1, 12}, {k, 1, n}]] (* A211799 *)
Table[k (k - 1) (2 k - 1)/6,
    {k, 1, z}] (* row-limit sequence, A000330 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A225923 Expansion of q^(-1/2) * k(q) * (1 - k(q)^4) * (K(q) / (Pi/2))^6 / 4 in powers of q where k(), k'(), K() are Jacobi elliptic functions.

Original entry on oeis.org

1, 20, -74, -24, 157, 124, 478, -1480, -1198, 3044, -480, 184, 2351, -1720, -3282, -5728, 2480, 1776, 10326, 9560, -8886, -9188, -11618, 23664, -16231, -23960, 11686, -9176, 60880, 16876, -18482, -3768, -35372, -15532, 3680, -31960, -4886, 47020, -2976, 44560
Offset: 0

Views

Author

Michael Somos, May 20 2013

Keywords

Comments

In Glaisher (1907) denoted by gamma(m) defined in section 63 on page 38.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In Chan and Combes (2024) page 2 is g(z) = eta(z)^8 eta(4z)^4 + 8 eta(4z)^12 identified as the unique newform of weight 6 and level 8 with LMFDB label 8.6.a.a. - Michael Somos, Jun 25 2025

Examples

			G.f. = 1 + 20*x - 74*x^2 - 24*x^3 + 157*x^4 + 124*x^5 + 478*x^6 - 1480*x^7 + ...
G.f. = q + 20*q^3 - 74*q^5 - 24*q^7 + 157*q^9 + 124*q^11 + 478*q^13 - 1480*q^15 + ...

References

  • J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 38).

Links

Crossrefs

Cf. A000735, A002292, A225872.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q]^12 + 32 q (QPochhammer[ q] QPochhammer[ q^4]^2)^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ 8 q QPochhammer[ q^4]^12 + (QPochhammer[ q]^2 QPochhammer[ q^4])^4, {q, 0, 2 n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^12 + 32 * x * eta(x + A)^4 * eta(x^4 + A)^8, n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( 8 * x * eta(x^4 + A)^12 + eta(x + A)^8 * eta(x^4 + A)^4, n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^6 / (eta(x + A) * eta(x^4 + A)^2))^4 + 16 * x * (eta(x + A) * eta(x^4 + A)^2)^4, n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2  + A)^12 / (eta(x + A)^5 * eta(x^4 + A)^4))^4 - x^2 * (4 * eta(x^4 + A)^4 / eta(x + A))^4, n))}; /* Michael Somos, Jul 20 2013 */

Formula

Expansion of (psi(x) * phi(-x^2)^2)^4 + 16 * x * (psi(x) * psi(-x)^2)^4 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (phi(x)^8 - 256 * x^2 * psi(x^2)^8) * psi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jul 20 2013
Expansion of q^(-1/2) * (eta(q)^12 + 32 * q * eta(q)^4 * eta(q^4)^8) in powers of q.
Expansion of q^(-1) * eta(q^4)^4 * (eta(q)^8 + 8 * eta(q^4)^8) in power of q^2. - Michael Somos, Jun 25 2025
G.f. is a period 1 Fourier series which satisfies f(-1/(8*t)) = 512 * (t/i)^6 * f(t) where q = exp(2*Pi*i*t).
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)) if p > 2.
G.f.: Product_{k>0} (1 - x^k)^12 + 32 * x * (Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2)^4.
|a(n)| = A002292(n). a(n) = A000735(n) + 32 * A225872(n).
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