|
| |
|
|
A119952
|
|
Number of partitions of n into parts relatively prime to 63 and not == 2 (mod 4).
|
|
2
|
|
|
|
1, 1, 1, 2, 3, 4, 5, 6, 9, 11, 13, 17, 22, 27, 32, 40, 51, 61, 72, 88, 108, 128, 150, 180, 217, 255, 297, 351, 416, 485, 562, 657, 770, 891, 1026, 1190, 1380, 1587, 1818, 2092, 2409, 2754, 3139, 3590, 4105, 4670, 5299, 6026, 6854, 7761, 8770, 9926, 11231
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
|
|
|
REFERENCES
|
B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 426 Entry 19(ii).
|
|
|
LINKS
|
|
|
|
FORMULA
|
Expansion of psi(-x^7) * psi(-x^9) / (psi(-x) * psi(-x^63)) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^6 * eta(q^2) * eta(q^7) * eta(q^9) * eta(q^28) * eta(q^36) * eta(q^126) / (eta(q) * eta(q^4) * eta(q^14) * eta(q^18) * eta(q^63) * eta(q^252)) in powers of q.
Euler transform of period 252 sequence A209198.
G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = f(t) where q = exp(2 Pi i t).
a(n) ~ exp(2*Pi*sqrt(2*n/21)) / (2^(3/4) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 29 2019
|
|
|
EXAMPLE
|
G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 9*x^8 + 11*x^9 + ...
G.f. = q^-6 + q^-5 + q^-4 + 2*q^-3 + 3*q^-2 + 4*q^-1 + 5 + 6*q + 9*q^2 + 11*q^3 + ...
a(7) = 6 since 5 + 1 + 1 = 4 + 3 = 4 + 1 + 1 + 1 = 3 + 3 + 1 = 3 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 in 6 ways.
|
|
|
MATHEMATICA
|
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^6*eta[q^2]* eta[q^7]*eta[q^9]*eta[q^28]*eta[q^36]*eta[q^126]/(eta[q]*eta[q^4]* eta[q^14]*eta[q^18]*eta[q^63]*eta[q^252]), {q, 0, 100}], q] (* G. C. Greubel, Apr 18 2018 *)
nmax = 100; CoefficientList[Series[Product[(1 - x^(7*k)) * (1 - x^(9*k)) * (1 + x^k) * (1 + x^(14*k)) * (1 + x^(18*k)) / ((1 - x^(4*k)) * (1 - x^(63*k)) * (1 + x^(126*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2019 *)
|
|
|
PROG
|
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^7 + A) * eta(x^9 + A) * eta(x^28 + A) * eta(x^36 + A) * eta(x^126 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^14 + A) * eta(x^18 + A) * eta(x^63 + A) * eta(x^252 + A)), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = sum( k=0, (sqrtint(8*n + 1) - 1)\2, (-x)^((k^2 + k)/2), x * O(x^n)); polcoeff( subst(A + x * O(x^(n\7)), x, x^7) * subst(A + x * O(x^(n\7)), x, x^9) / A / subst(A + x * O(x^(n\63)), x, x^63), n))};
(PARI) my(q='q+O('q^99)); Vec(eta(q^2)*eta(q^7)*eta(q^9)*eta(q^28)*eta(q^36)*eta(q^126)/(eta(q)*eta(q^4)*eta(q^14)*eta(q^18)*eta(q^63)*eta(q^252))) \\ Altug Alkan, Apr 18 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
