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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A036026 Number of partitions of n into parts not of forms 4*k+2, 20*k, 20*k+5 or 20*k+15.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 6, 8, 10, 12, 16, 21, 25, 30, 38, 48, 57, 68, 84, 102, 121, 143, 172, 207, 243, 284, 338, 400, 465, 542, 636, 744, 862, 996, 1158, 1344, 1546, 1776, 2050, 2361, 2701, 3088, 3540, 4050, 4613, 5248, 5980, 6808, 7719, 8742, 9916, 11232
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Case k=5,i=3 of Gordon/Goellnitz/Andrews Theorem.
Number of partitions in which no odd part is repeated, with at most 2 parts of size less than or equal to 2 and where differences between parts at distance 4 are greater than 1 when the smaller part is odd and greater than 2 when the smaller part is even.

Examples

			1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...
q + q^3 + q^5 + 2*q^7 + 3*q^9 + 3*q^11 + 4*q^13 + 6*q^15 + 8*q^17 + ...

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.

Links

Crossrefs

Cf. A101277, A145708, A145722.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[EllipticTheta[2, 0, I*q^(5/2)]/( q^(1/2) * EllipticTheta[2, 0, I*q^(1/2)]), {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Apr 16 2017 *)
nmax = 60; CoefficientList[Series[Product[(1 - x^(4*k - 2))*(1 - x^(20*k))*(1 - x^(20*k+5-20))*(1 - x^(20*k- 5))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))} /* Michael Somos, Oct 27 2008 */

Formula

Expansion of psi(-x^5) / psi(-x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Oct 27 2008
Expansion of q^(-1/2) * eta(q^2) * eta(q^5) * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10)) in powers of q. - Michael Somos, Oct 27 2008
Euler transform of period 20 sequence [ 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, ...]. - Michael Somos, Oct 27 2008
a(2*n) = A145722(n). a(2*n + 1) = A101277(n). Convolution inverse of A145708.
a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(5/4) * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, May 10 2018

A201076 Irregular triangle read by rows: number of {0,2}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 3, 6, 7, 7, 5, 0, 0, 1, 2, 3, 6, 10, 13, 16, 20, 19, 15, 8, 0, 0, 1, 2, 3, 6, 10, 16, 22, 29, 39, 48, 53, 56, 57, 46, 30, 13, 0, 0, 0, 1, 2, 3, 6, 10, 16, 25, 35, 48, 66, 85, 106, 127, 147, 167, 179, 178, 168, 146, 103, 58, 21, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane, Nov 26 2011

Keywords

Examples

			Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 3 6 7 7 5 0 0
1 2 3 6 10 13 16 20 19 15 8 0 0
...

Links

Crossrefs

Row sums give A052709. Rows converge to A101277.
Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201079 {0,2,4,6...}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

Programs

  • Mathematica
    gf = Expand /@ FixedPoint[1 + (q x + q^2 x^2) # (Normal@# /. {x :> q^2 x}) + O[x]^8 &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Extensions

Row 5 corrected, rows 6-7 added by Andrey Zabolotskiy, Jan 02 2024
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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