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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)
41
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).

Convolution of A000009 and A035457. - Vaclav Kotesovec, Aug 23 2015

REFERENCES

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.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.2).

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

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, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129.]

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, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).

M. D. Hirschhorn, J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.

Alexander Patkowski, On some partitions where even parts do not repeat, Demonstratio Mathematica Volume 42, Issue 2 (Jun 2009), pp. 259-263.

Eric Weisstein's World of Mathematics, Partition Function b_k and 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) * Product_{k=1..n} (1+x^k)/(1-x^k) ). - Joerg Arndt, Apr 07 2011

G.f.: P(x^4)/P(x) where P(x) = Product_{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

a(n) ~ exp(Pi*sqrt(n/2)) / (4 * (2*n)^(3/4)). - Vaclav Kotesovec, Aug 23 2015

EXAMPLE

G.f. = 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 + ...

G.f. = 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

# second Maple program:

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(

     `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Nov 24 2015

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-François Alcover, May 26 2011, after Jon Perry *)

QP = QPochhammer; CoefficientList[QP[q^4]/QP[q] + O[q]^50, q] (* Jean-François Alcover, Nov 24 2015 *)

a[0] = 1; a[n_] := a[n] = Sum[a[n-j] DivisorSum[j, If[Divisible[#, 4], 0, #]&], {j, 1, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)

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

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

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

Sequence in context: A271147 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 July 26 02:47 EDT 2016. Contains 275038 sequences.