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A001935
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Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4
(Formerly M0566 N0204)
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25
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 [From 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).
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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).
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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, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Partition Function P
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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 7 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).
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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].
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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
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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] (* From Jean-François Alcover, May 26 2011, after J. Perry *)
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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 */
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CROSSREFS
| Cf. A000009, A000726, A001936, A035959, A061198, A061199, A081055, A081056, A083365, A098491, A098492.
Cf. A000009, A000041, A010054 [From Gary W. Adamson, Mar 16 2010]
Cf. A174715 [From Gary W. Adamson, Mar 27 2010]
Sequence in context: A186115 A069907 * A083365 A007604 A013950 A018550
Adjacent sequences: A001932 A001933 A001934 * A001936 A001937 A001938
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu)
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