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A027193 Number of partitions of n into an odd number of parts. 21
0, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 29, 37, 52, 66, 90, 113, 151, 190, 248, 310, 400, 497, 632, 782, 985, 1212, 1512, 1851, 2291, 2793, 3431, 4163, 5084, 6142, 7456, 8972, 10836, 12989, 15613, 18646, 22316, 26561, 31659, 37556, 44601, 52743, 62416, 73593, 86809, 102064, 120025, 140736 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

Number of partitions of n+1 into an even number of parts, the least being 1. Example: a(5)=4 because we have [5,1], [3,1,1,1], [2,1,1] and [1,1,1,1,1,1].

Also number of partitions of n+1 such that the largest part is even and occurs only once. Example: a(5)=4 because we have [6], [4,2], [4,1,1] and [2,1,1,1,1]. - Emeric Deutsch, Apr 05 2006

Also the number of partitions of n such that the number of odd parts has the parity of n.  Example: a(8)=10 because we have 8, 611, 521, 431, 422, 41111, 332, 32111, 22211, 2111111. - Clark Kimberling, Feb 01 2014

In Chaves 2011 see page 38 equation (3.20). - Michael Somos, Dec 28 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; p. 39, Example 7.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..999

D. R. C. Chaves, Um estudo combinatório e comparativo de identidades teta parciais de Andrews e Ramanujan 2011

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_0(n).

FORMULA

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

For g.f. see under A027187.

G.f.: Sum(k>=1, x^(2*k-1)/Product(j=1..2*k-1, 1-x^j ) ). - Emeric Deutsch, Apr 05 2006

G.f.: - Sum(k>=1, (-x)^(k^2)) / Product(k>=1, 1-x^k ). - Joerg Arndt, Feb 02 2014

G.f.: Sum(k>=1, x^(k*(2*k-1)) / Product(j=1..2*k, 1-x^j)). - Michael Somos, Dec 28 2014

a(2*n) = A000701(2*n), a(2*n-1) = A046682(2*n-1); a(n) = A000041(n)-A027187(n). - Reinhard Zumkeller, Apr 22 2006

EXAMPLE

G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 10*x^8 + 16*x^9 + 20*x^10 + ...

MAPLE

g:=sum(x^(2*k)/product(1-x^j, j=1..2*k-1), k=1..40): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=1..45); # Emeric Deutsch, Apr 05 2006

MATHEMATICA

nn=40; CoefficientList[Series[ Sum[x^(2j+1)Product[1/(1- x^i), {i, 1, 2j+1}], {j, 0, nn}], {x, 0, nn}], x]  (* Geoffrey Critzer, Dec 01 2012 *)

a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n], OddQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)

a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n], OddQ[ First@#] &]]; (* Michael Somos, Dec 28 2014 *)

a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n + 1], #[[-1]] == 1 && EvenQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)

a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n + 1], EvenQ[ First@#] && (Length[#] < 2 || #[[1]] != #[[2]]) &]]; (* Michael Somos, Dec 28 2014 *)

PROG

(PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=1, n, if( k%2, x^k / prod( j=1, k, 1 - x^j, 1 + x * O(x^(n-k)) ))), n))}; /* Michael Somos, Jul 24 2012 */

(PARI) q='q+O('q^66); concat([0], Vec( (1/eta(q)-eta(q)/eta(q^2))/2 ) ) \\ Joerg Arndt, Mar 23 2014

CROSSREFS

Cf. A000041, A000700, A000701, A026837, A027187, A046682.

Sequence in context: A193146 A240486 A237871 * A126796 A240451 A206138

Adjacent sequences:  A027190 A027191 A027192 * A027194 A027195 A027196

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified July 22 18:10 EDT 2017. Contains 289671 sequences.