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A115671
Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).
6
1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369
OFFSET
0,4
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Andrews (1987) refers to this sequence as p(S, n) where S is the set in equation (1) on page 437.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..120 from Reinhard Zumkeller)
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 32. MR0858826 (88b:11063).
G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.
Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157, December 2015, Pages 223-232.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (f(q) / f(-q) + 1) / 2 in powers of q where f() is a Ramanujan theta function.
Expansion of f(q^6, q^10) / f(-q, -q^3) = (f(q^22, q^26) - q^2 * f(q^10, q^38)) / f(-q, -q^2) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 32 sequence [ 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(x) = (2*A(x) - 1)^2 = g.f. A007096 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 1 + u^2 - 2 * u * v^2.
G.f. (1 + sqrt( theta_3(x) / theta_4(x))) / 2 = (Sum_{k} x^(8*k^2 - 2*k)) / (Sum_{k} (-x)^(2*k^2 - k)) = (Sum_{k} x^(24*n^2 + 2*n) - x^(24*n^2 + 14*n + 2)) / (Product_{k>0} 1 - x^k).
2 * a(n) = A080054(n) unless n = 0. a(2*n + 2) = A208851(n). a(2*n + 1) = A187154(n). a(n + 1) = A208856(n).
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 15*x^9 + ...
a(5) = 4 since 5 = 4 + 1 = 3 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 4 ways.
a(6) = 6 since 6 = 5 + 1 = 4 + 1 + 1 = 3 + 3 = 3 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 6 ways.
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / QPochhammer[ q]^2 / QPochhammer[ q^4] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A))) / 2, n))};
(Haskell)
a115671 = p [x | x <- [0..], (mod x 32) `notElem` [0, 2, 12, 14, 16, 18, 20, 30]]
where p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Mar 03 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 29 2006
STATUS
approved