A115671 - OEIS

%I #34 Feb 16 2025 08:33:00

%S 1,1,1,2,3,4,6,8,11,15,20,26,34,44,56,72,91,114,143,178,220,272,334,

%T 408,498,605,732,884,1064,1276,1528,1824,2171,2580,3058,3616,4269,

%U 5028,5910,6936,8124,9498,11088,12922,15034,17468,20264,23472,27154,31369

%N Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Andrews (1987) refers to this sequence as p(S, n) where S is the set in equation (1) on page 437.

%H G. C. Greubel, <a href="/A115671/b115671.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..120 from Reinhard Zumkeller)

%H G. E. Andrews, <a href="https://bookstore.ams.org/cbms-66/">q-series</a>, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 32. MR0858826 (88b:11063).

%H G. E. Andrews, <a href="http://www.jstor.org/stable/2322727">Unsolved Problems: Further Problems on Partitions</a>, Amer. Math. Monthly 94 (1987), no. 5, 437-439.

%H Mircea Merca, <a href="https://doi.org/10.1016/j.jnt.2015.05.008">The bisectional pentagonal number theorem</a>, Journal of Number Theory, Volume 157, December 2015, Pages 223-232.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (f(q) / f(-q) + 1) / 2 in powers of q where f() is a Ramanujan theta function.

%F 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.

%F 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, ...].

%F 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.

%F 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).

%F 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).

%e 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 + ...

%e a(5) = 4 since 5 = 4 + 1 = 3 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 4 ways.

%e 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.

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] + 1) / 2, {q, 0, n}]; (* _Michael Somos_, Nov 09 2014 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / QPochhammer[ q]^2 / QPochhammer[ q^4] + 1) / 2, {q, 0, n}]; (* _Michael Somos_, Nov 09 2014 *)

%o (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))};

%o (Haskell)

%o a115671 = p [x | x <- [0..], (mod x 32) `notElem` [0,2,12,14,16,18,20,30]]

%o    where p _          0 = 1

%o          p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Mar 03 2012

%Y Cf. A080054, A187154, A208851, A208856, A218171.

%K nonn,changed

%O 0,4

%A _Michael Somos_, Jan 29 2006