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A027187 Number of partitions of n into an even number of parts. 24
1, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 40, 49, 69, 86, 118, 146, 195, 242, 317, 392, 505, 623, 793, 973, 1224, 1498, 1867, 2274, 2811, 3411, 4186, 5059, 6168, 7427, 9005, 10801, 13026, 15572, 18692, 22267, 26613, 31602, 37619, 44533, 52815, 62338, 73680, 86716, 102162, 119918 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

Number of partitions of n in which greatest part is even.

Number of partitions of n+1 into an odd number of parts, the least being 1.

Also the number of partitions of n such that the number of even parts has the parity of n; see Comments at A027193. - Clark Kimberling, Feb 01 2014

Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

REFERENCES

Roland Bacher, P De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016. hal-01285685v2; https://hal.archives-ouvertes.fr/hal-01285685/document

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.

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 p_e(n).

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

a(n) = (A000041(n) + (-1)^n * A000700(n))/2.

a(n)=p(n)-p(n-1)+p(n-4)-p(n-9)+... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan, Mar 14 2004

G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3 q^4 + 3 q^5 + 6 q^6 + ...

= Sum_{n >= 0} q^(2n)/(q; q)_{2n}

= ((Prod_{k >= 1} 1/(1-q^k) + (Prod_{k >= 1} 1/(1+q^k))/2 [Bill Gosper, Jun 25 2005]

Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2 q^3 + 2 q^4 + 4 q^5 + 5 q^6 + ...

Then B(q) = Sum_{n >= 0} q^(2n+1)/(q; q)_{2n+1} = ((Prod_{k >= 1} 1/(1-q^k) - (Prod_{k >= 1} 1/(1+q^k))/2.

Also we have the following identity involving 2 X 2 matrices:

Prod_{k >= 1} [ 1/(1-q^2k) q^k/(1-q^2k / q^k/(1-q^2k) 1/(1-q^2k) ]

= [ A(q) B(q) / B(q) A(q) ] [Bill Gosper, Jun 25 2005]

a(2*n) = A046682(2*n), a(2*n+1) = A000701(2*n+1); a(n) = A000041(n)-A027193(n). - Reinhard Zumkeller, Apr 22 2006

Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - Michael Somos, Aug 19 2006

G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - Michael Somos, Aug 19 2006

EXAMPLE

G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...

MATHEMATICA

f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* Clark Kimberling, Mar 13 2012 *)

a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)

a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* Michael Somos, May 06 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* Michael Somos, Aug 19 2006 */

(PARI) q='q+O('q^66); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ Joerg Arndt, Mar 23 2014

CROSSREFS

Cf. A027193, A000701, A046682, A026838, A102462.

Sequence in context: A240449 A088571 A241832 * A056508 A050065 A078477

Adjacent sequences:  A027184 A027185 A027186 * A027188 A027189 A027190

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Offset changed to 0 by Michael Somos, Jul 24 2012

STATUS

approved

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Last modified March 25 00:01 EDT 2017. Contains 284035 sequences.