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 A067661 Number of partitions of n into distinct parts such that number of parts is even. 23
 1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3959, 4404 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Ramanujan theta functions: phi(q) (A000122), chi(q) (A000700). REFERENCES B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (2). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q_e(n). Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^3 + q^4 + 2 q^5 + 2 q^6 + 3 q^7 + ... = Sum_{n >= 0} q^(n(2n+1))/(q; q)_{2n} [Bill Gosper, Jun 25 2005] Also, let B(q) = Sum_{n >= 0} A067659(n) q^n = q + q^2 + q^3 + q^4 + q^5 + 2 q^6 + ... Then B(q) = Sum_{n >= 0} q^((n+1)(2n+1))/(q; q)_{2n+1}. Also we have the following identity involving 2 X 2 matrices: Prod_{k >= 1} [ 1 q^k / q^k 1 ] = [ A(q) B(q) / B(q) A(q) ] [Bill Gosper, Jun 25 2005] a(n) = (A000009(n)+A010815(n))/2. - Vladeta Jovovic, Feb 24 2002 Expansion of (1 + phi(-x)) / (2*chi(-x)) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006 a(n) + A067659(n) = A000009(n). - R. J. Mathar, Jun 18 2016 a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 24 2018 EXAMPLE G.f. = 1 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + ... MAPLE b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,       `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))     end: a:= n-> b(n\$2, 1): seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014 MATHEMATICA b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *) a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] + QPochhammer[ x]) / 2, {x, 0, n}]; (* Michael Somos, May 06 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A) + eta(x + A)) / 2, n))}; /* Michael Somos, Feb 14 2006 */ (PARI) N=66;  q='q+O('q^N);  S=1+2*sqrtint(N); gf=sum(n=0, S, (n%2==0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ); Vec(gf)  \\ Joerg Arndt, Apr 01 2014 CROSSREFS Cf. A067659 (odd number of parts). Sequence in context: A026798 A185325 A125890 * A210024 A052839 A125894 Adjacent sequences:  A067658 A067659 A067660 * A067662 A067663 A067664 KEYWORD easy,nonn AUTHOR Naohiro Nomoto, Feb 23 2002 STATUS approved

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Last modified July 20 13:55 EDT 2019. Contains 325181 sequences. (Running on oeis4.)