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A067661 Number of partitions of n into distinct parts such that number of parts is even. 22
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).

M. Somos, Introduction to Ramanujan theta functions

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-Fran├žois 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 August 20 00:52 EDT 2018. Contains 313902 sequences. (Running on oeis4.)