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A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.
(Formerly M0566 N0204)
33
1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, 105, 132, 166, 208, 258, 320, 395, 484, 592, 722, 876, 1060, 1280, 1539, 1846, 2210, 2636, 3138, 3728, 4416, 5222, 6163, 7256, 8528, 10006, 11716, 13696, 15986, 18624, 21666, 25169, 29190, 33808, 39104, 45164 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also number of partitions of n where no part appears more than three times.

a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al).

Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - Emeric Deutsch, Apr 19 2006

Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - Gary W. Adamson, Mar 16 2010

Equals left border of triangle A174715. - Gary W. Adamson, Mar 27 2010

The Cayley reference is actually to A083365. - Michael Somos, Feb 24 2011

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

REFERENCES

G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 9.)

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

S.-C. Chen, On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943.

A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241.

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

Joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?

Eric Weisstein's World of Mathematics, Partition Function b_k.

Eric Weisstein's World of Mathematics, Partition Function P

FORMULA

Euler transform of period 4 sequence [ 1, 1, 1, 0, ...].

Expansion of q^(-1/8)*eta(q^4)/eta(q) in powers of q. - Michael Somos, Mar 19 2004

Expansion of psi(-x) / phi(-x) = psi(x) / phi(-x^2) = psi(x^2) / psi(-x) = chi(x) / chi(-x^2)^2 = 1 / (chi(x) * chi(-x)^2) = 1 / (chi(-x) * chi(-x^2)) = f(-x^4) / f(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 08 2011

G.f.: Product(j>=1, 1 + x^j + x^(2*j) + x^(3*j)). - Jon Perry, Mar 30 2004

G.f.: product(k>=1, (1+x^k)^(2-k%2)). - Jon Perry, May 05 2005

G.f.: Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)) = 1 + Sum_{k>0}(Product_{i=1..k} (x^i + 1) / (x^-i - 1)).

G.f.: sum(n>=0, x^(n*(n+1)/2) * prod(k=1..n, (1+x^k)/(1-x^k) ) ). - Joerg Arndt, Apr 07 2011

G.f.: P(x^4)/P(x) where P(x)=prod(k>=1, 1-x^k ). - Joerg Arndt, Jun 21 2011

A083365(n) = (-1)^n a(n). Convolution square is A001936. a(n) = A098491(n) + A098492(n). a(2*n) = A081055(n). a(2*n + 1) = A081056(n).

G.f.:  (1+ 1/G(0))/2, where G(k)= 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013

G.f.: exp( Sum_{n>=1} (x^n/n) / (1 + (-x)^n) ). - Paul D. Hanna, Jul 24 2013

EXAMPLE

1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 16*x^8 + 22*x^9 + ...

q + q^9 + 2*q^17 + 3*q^25 + 4*q^33 + 6*q^41 + 9*q^49 + 12*q^57 + 16*q^65 + 22*q^73 + ...

a(5)=6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1].

MAPLE

g:=product((1+x^j)*(1+x^(2*j)), j=1..50): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..48); - Emeric Deutsch, Apr 19 2006

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8), {q, 0, n}] (* Michael Somos, Jul 11 2011 *)

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}], {x, 0, n}] (* Michael Somos, Jul 08 2011 *)

CoefficientList[Series[Product[1+x^j+x^(2j)+x^(3j), {j, 1, 48}], {x, 0, 48}], x] (* Jean-Fran├žois Alcover, May 26 2011, after Jon Perry *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)), n))}

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 8*n + 1) - 1)\2, prod(i=1, k, (1 + x^i) / (x^-i - 1), 1 + x * O(x^n))), n))} /* Michael Somos, Jun 01 2004 */

(Haskell)

a001935 = p a042968_list where

   p _          0 = 1

   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

-- Reinhard Zumkeller, Sep 02 2012

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(1+(-x)^m+x*O(x^n))/m)), n)} \\ Paul D. Hanna, Jul 24 2013

CROSSREFS

Cf. A000009, A000726, A035959, A219601, A035985, A001936, A061198, A061199, A081055, A081056, A083365, A098491, A098492, A042968, A070048.

Cf. A000041, A010054. - Gary W. Adamson, Mar 16 2010

Cf. A174715. - Gary W. Adamson, Mar 27 2010

Sequence in context: A186115 A069907 A083365 * A007604 A013950 A018550

Adjacent sequences:  A001932 A001933 A001934 * A001936 A001937 A001938

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe, Robert G. Wilson v

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

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Last modified April 17 14:46 EDT 2014. Contains 240646 sequences.