%I #14 Mar 12 2021 22:24:43

%S 1,0,0,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1,0,0,2,0,0,1,

%T 1,0,0,0,0,0,1,0,2,1,0,1,0,0,0,2,0,0,0,0,1,0,0,1,1,0,1,0,0,2,0,0,0,1,

%U 0,2,1,0,0,0,0,1,0,0,1,0,0,0,0,0,2,1,0,2,1,0,1,0,0,1,0,0,1,0,0,0,0,0,1,1,0

%N Number of representations of n as a sum of three times a triangular number and four times a triangular number.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%D M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

%H G. C. Greubel, <a href="/A112609/b112609.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F a(n) = 1/2*( d_{1, 3}(8n+7) - d_{2, 3}(8n+7) ) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.

%F Expansion of phi(q^3) * psi(q^4) in powers of q where psi() is a Ramanujan theta function. - _Michael Somos_, Mar 10 2008

%F Expansion of q^(-7/8) * (eta(q^6) * eta(q^8))^2 / (eta(q^3) * eta(q^4)) in powers of q. - _Michael Somos_, Mar 10 2008

%F Euler transform of period 24 sequence [ 0, 0, 1, 1, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 0, -2, ...]. - _Michael Somos_, Mar 10 2008

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138270.

%F a(3*n+2) = 0.

%e a(30) = 2 since we can write 30 = 3*10 + 4*0 = 3*6 + 4*3

%e q^7 + q^31 + q^39 + q^63 + q^79 + q^103 + q^111 + q^127 + q^151 + ...

%t A112609[n_] := SeriesCoefficient[(QPochhammer[q^6]*QPochhammer[q^8])^2/

%t (QPochhammer[q^3]*QPochhammer[q^4]), {q,0,n}]; Table[A112609[n], {n, 0, 50}] (* _G. C. Greubel_, Sep 25 2017 *)

%o (PARI) {a(n) = if( n<0, 0, n=8*n+7; sumdiv(n, d, kronecker(-3, d))/2)} /* _Michael Somos_, Mar 10 2008 */

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^8 + A))^2 / (eta(x^3 + A) * eta(x^4 + A)), n))} /* _Michael Somos_, Mar 10 2008 */

%Y A131962(n) = a(3*n). A112607(n) = a(3*n+1). A128617(n) = a(4*n+3).

%Y A112605(2*n+1) = 2 * a(n). A112607(3*n+1) = a(n). A033762(4*n+3) = 2 * a(n). A112604(6*n+5) = 2 * a(n). A002324(8*n+7) = a(n). A123484(24*n+21) = 2 * a(n).

%K nonn

%O 0,31

%A _James A. Sellers_, Dec 21 2005