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A000701 One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.
(Formerly M0645 N0239)
18
0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 190, 242, 310, 392, 497, 623, 782, 973, 1212, 1498, 1851, 2274, 2793, 3411, 4163, 5059, 6142, 7427, 8972, 10801, 12989, 15572, 18646, 22267, 26561, 31602, 37556, 44533, 52743, 62338, 73593 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Also number of cycle types of odd permutations.

Also number of partitions of n with an odd number of even parts. There is no restriction on the odd parts. - N. Sato, Jul 20 2005. E.g., a(6)=5 because we have [6],[4,1,1],[3,2,1],[2,2,2] and [2,1,1,1,1]. - Emeric Deutsch, Mar 02 2006

Also number of partitions of n with largest part not congruent to n modulo 2: a(2*n)=A027193(2*n), a(2*n+1)=A027187(2*n+1); a(n)=A000041(n)-A046682(n). - Reinhard Zumkeller, Apr 22 2006

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.

FORMULA

a(n) = (A000041(n) - A000700(n))/2.

From Bill Gosper, Aug 08 2005: (Start)

Sum a(n) q^n = q^2 + q^3 + 2 q^4 + 3 q^5 + 5 q^6 + 7 q^7 + ...

= -( Sum_{n = 1 .. oo} (-q^2)^(n^2) ) / ( Sum_{ n = -oo, oo } (-1)^n q^(n(3n-1)/2) )

= (- q; q)_{oo} Sum_{n=1..oo} q^(2(2n-1))/(q^2;q^2)_{2n-1}

= (1/(q;q)_oo - 1/(q;-q)_oo)/2

= (1/(q;q)_oo - (-q;q^2)_oo)/2

= Sum{ k = 0..oo } ( 1/((q;q)_k)^2 - 1/(q^2;q^2)_k ) q^(k^2)/2

using the "q-pochhammer" notation (a;q)_n := Product_{k=0..n-1} 1-a*q^k.

(End)

a(n) = p(n-2) - p(n-8) + p(n-18) - p(n-32) + ... + (-1)^(k+1)*p(n-2*k^2) + ..., where p() is A000041(). E.g., a(20) = p(18) - p(12) + p(2) = 385 - 77 + 2 = 310. - Vladeta Jovovic, Aug 08 2004

G.f. = (1/2)(1-product((1-x^(2j))/(1+x^(2j)), j=1..infinity))/product(1-x^j, j=1..infinity). - Emeric Deutsch, Mar 02 2006

a(2*n) = A236559(n). a(2*n + 1) = A236914(n). - Michael Somos, Aug 25 2015

EXAMPLE

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

MAPLE

with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2;

MATHEMATICA

a41 = PartitionsP; a700[n_] := SeriesCoefficient[ Product[1 + x^k, {k, 1, n, 2}], {x, 0, n}]; a[0] = 0; a[n_] := (a41[n] - a700[n])/2; Table[a[n], {n, 0, 48}] (* Jean-Fran├žois Alcover, Feb 21 2012, after first formula *)

a[ n_] := SeriesCoefficient[ (1 / QPochhammer[ x] - 1 / QPochhammer[ x, -x]) / 2, {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)

a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x^2]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] Sum[ x^(2 k) / QPochhammer[ x^2, x^2, k], {k, 1, n/2, 2}], {x, 0, n}] (* Michael Somos, Aug 25 2015 *)

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (1 / QPochhammer[ x, x, k]^2 - 1 / QPochhammer[ x^2, x^2, k]) x^k^2, {k, Sqrt@n}] / 2, {x, 0, n}]]; (* Michael Somos, Aug 25 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x^2 + A)^2 / eta(x^4 + A) ) / (2 * eta(x + A)), n))}; /* Michael Somos, Aug 25 2015 */

CROSSREFS

Cf. A000041, A000700, A027187, A027193, A046682, A236559, A236914.

Cf. A118302.

Sequence in context: A036005 A104503 A027340 * A123975 A214077 A094984

Adjacent sequences:  A000698 A000699 A000700 * A000702 A000703 A000704

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description and more terms from Christian G. Bower, Apr 27 2000

STATUS

approved

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Last modified March 26 10:42 EDT 2017. Contains 284111 sequences.