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A027187
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Number of partitions of n into an even number of parts.
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185
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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
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OFFSET
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0,5
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COMMENTS
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For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]
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 same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
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
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.
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LINKS
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N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
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FORMULA
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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}
= ((Product_{k >= 1} 1/(1-q^k) + (Product_{k >= 1} 1/(1+q^k))/2.
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} = ((Product_{k >= 1} 1/(1-q^k) - (Product_{k >= 1} 1/(1+q^k))/2.
Also we have the following identity involving 2 X 2 matrices:
Product_{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) ]. (End)
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
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EXAMPLE
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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 + ...
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.
(11) (21) (22) (32) (33) (43) (44)
(31) (41) (42) (52) (53)
(1111) (2111) (51) (61) (62)
(2211) (2221) (71)
(3111) (3211) (2222)
(111111) (4111) (3221)
(211111) (3311)
(4211)
(5111)
(221111)
(311111)
(11111111)
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.
(2) (21) (4) (41) (6) (43) (8)
(22) (221) (42) (61) (44)
(211) (2111) (222) (421) (62)
(411) (2221) (422)
(2211) (4111) (431)
(21111) (22111) (611)
(211111) (2222)
(4211)
(22211)
(41111)
(221111)
(2111111)
(End)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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The Heinz numbers of these partitions are A028260.
The case of even sum as well as length is A236913 (the even bisection).
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A026805 counts partitions whose least part is even.
A072233 counts partitions by sum and length.
A101708 counts partitions of even positive rank.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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