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

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

0,4

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.

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A101277, A145708, A145722.

Sequence in context: A036025 A036030 A036022 * A116494 A036031 A218947

Adjacent sequences: A036023 A036024 A036025 * A036027 A036028 A036029

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved