|
%I #92 Jun 14 2024 11:41:30
%S 1,0,1,1,3,3,6,7,12,14,22,27,40,49,69,86,118,146,195,242,317,392,505,
%T 623,793,973,1224,1498,1867,2274,2811,3411,4186,5059,6168,7427,9005,
%U 10801,13026,15572,18692,22267,26613,31602,37619,44533,52815,62338,73680,86716,102162,119918
%N Number of partitions of n into an even number of parts.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C For n > 0, also the number of partitions of n whose greatest part is even. [Edited by _Gus Wiseman_, Jan 05 2021]
%C Number of partitions of n+1 into an odd number of parts, the least being 1.
%C Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - _Clark Kimberling_, Feb 01 2014, corrected Jan 06 2021
%C 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
%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.
%H Alois P. Heinz, <a href="/A027187/b027187.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)
%H George E. Andrews and David Newman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Andrews/andrews5.html">The Minimal Excludant in Integer Partitions</a>, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
%H Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, <a href="https://arxiv.org/abs/2406.06036">How large is the character degree sum compared to the character table sum for a finite group?</a>, arXiv:2406.06036 [math.RT], 2024. See p. 13.
%H Roland Bacher and Pierre De La Harpe, <a href="https://hal.science/hal-01285685">Conjugacy growth series of some infinitely generated groups</a>, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
%H N. J. Fine, <a href="http://www.jstor.org/stable/2307653">Problem 4314</a>, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
%H Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).
%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) = (A000041(n) + (-1)^n * A000700(n))/2.
%F 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
%F From _Bill Gosper_, Jun 25 2005: (Start)
%F 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 + ...
%F = Sum_{n >= 0} q^(2n)/(q; q)_{2n}
%F = ((Product_{k >= 1} 1/(1-q^k) + (Product_{k >= 1} 1/(1+q^k))/2.
%F 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 + ...
%F Then B(q) = Sum_{n >= 0} q^(2n+1)/(q; q)_{2n+1} = ((Product_{k >= 1} 1/(1-q^k) - (Product_{k >= 1} 1/(1+q^k))/2.
%F Also we have the following identity involving 2 X 2 matrices:
%F Product_{k >= 1} [ 1/(1-q^2k) q^k/(1-q^2k / q^k/(1-q^2k) 1/(1-q^2k) ]
%F = [ A(q) B(q) / B(q) A(q) ]. (End)
%F 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
%F Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - _Michael Somos_, Aug 19 2006
%F G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - _Michael Somos_, Aug 19 2006
%F a(n) = A338914(n) + A096373(n). - _Gus Wiseman_, Jan 06 2021
%e 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 + ...
%e From _Gus Wiseman_, Jan 05 2021: (Start)
%e The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by A028260.
%e (11) (21) (22) (32) (33) (43) (44)
%e (31) (41) (42) (52) (53)
%e (1111) (2111) (51) (61) (62)
%e (2211) (2221) (71)
%e (3111) (3211) (2222)
%e (111111) (4111) (3221)
%e (211111) (3311)
%e (4211)
%e (5111)
%e (221111)
%e (311111)
%e (11111111)
%e The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by A244990.
%e (2) (21) (4) (41) (6) (43) (8)
%e (22) (221) (42) (61) (44)
%e (211) (2111) (222) (421) (62)
%e (411) (2221) (422)
%e (2211) (4111) (431)
%e (21111) (22111) (611)
%e (211111) (2222)
%e (4211)
%e (22211)
%e (41111)
%e (221111)
%e (2111111)
%e (End)
%t f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* _Clark Kimberling_, Mar 13 2012 *)
%t a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* _Michael Somos_, May 06 2015 *)
%t a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* _Michael Somos_, May 06 2015 *)
%o (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 */
%o (PARI) q='q+O('q^66); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ _Joerg Arndt_, Mar 23 2014
%Y Cf. A000701, A046682, A026838, A102462.
%Y The Heinz numbers of these partitions are A028260.
%Y The odd version is A027193.
%Y The strict case is A067661.
%Y The case of even sum as well as length is A236913 (the even bisection).
%Y Other cases of even length:
%Y - A024430 counts set partitions of even length.
%Y - A034008 counts compositions of even length.
%Y - A052841 counts ordered set partitions of even length.
%Y - A174725 counts ordered factorizations of even length.
%Y - A332305 counts strict compositions of even length
%Y - A339846 counts factorizations of even length.
%Y A000009 counts partitions into odd parts, ranked by A066208.
%Y A026805 counts partitions whose least part is even.
%Y A072233 counts partitions by sum and length.
%Y A101708 counts partitions of even positive rank.
%Y Cf. A000700, A026424, A058696, A096373, A244990, A300061.
%K nonn
%O 0,5
%A _Clark Kimberling_
%E Offset changed to 0 by _Michael Somos_, Jul 24 2012
| 