0,3

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

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

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 (deutsch(AT)duke.poly.edu), Apr 19 2006

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

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

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.

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.

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Partition Function P

G.f.: Product(j=1, oo, 1 + x^j + x^2j + x^3j) - Jon Perry (perry(AT)globalnet.co.uk), Mar 30 2004

G.f.: product(k=1, oo, (1+x^k)^(2-k%2)) - Jon Perry (perry(AT)globalnet.co.uk), May 05 2005

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

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

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 (deutsch(AT)duke.poly.edu), Apr 19 2006

(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 */

A083365(n)=(-1)^n a(n). Convolution square is A001936. Cf. A000009, A000726, A035959, A061198, A061199.

Equals A098491 + A098492.

Adjacent sequences: A001932 A001933 A001934 this_sequence A001936 A001937 A001938

Sequence in context: A058647 A073576 A069907 this_sequence A083365 A007604 A013950

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, Robert G. Wilson v (rgwv(AT)rgwv.com)

More terms from James A. Sellers (sellersj(AT)math.psu.edu)