0,7

Expansion of Rogers-Ramanujan function H(x) in powers of x.

Also number of partitions of n such that the number of parts is greater by one than the smallest part. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 04 2006

Example: a(10)=4 because we have [9, 1], [6, 2, 2], [5, 3, 2] and [4, 4, 2]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2006

Also number of partitions of n such that if the largest part is k, then there are exactly k-1 parts equal to k. Example: a(10)=4 because we have [3, 3, 2, 2], [3, 3, 2, 1, 1], [3, 3, 1, 1, 1, 1] and [2, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2006

Also number of partitions of n such that if the largest part is k, then k occurs at least k+1 times. Example: a(10)=4 because we have [2, 2, 2, 2, 2], [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2006

Also number of partitions of n such that the smallest part is larger than the number of parts. Example: a(10)=4 because we have [10], [7, 3], [6, 4] and [5, 5]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2006

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

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 238.

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

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(f), p. 591.

G. E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities. Amer. Math. Monthly 96 (1989), no. 5, 401-409.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

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

P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums

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

The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n*(n+1))/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-2))*(1-t^(5*n-3))); this is the g.f. for the sequence.

G.f.: (Product_{k>0} 1+x^(2k))*(Sum_{k>=0} x^(k^2+2k)/(Product_{i=1..k} 1-x^(4i))). - Michael Somos Oct 19 2006

Euler transform of period 5 sequence [ 0, 1, 1, 0, 0, ...]. - Michael Somos Oct 15 2008

a(10)=4 because we have [8, 2], [7, 3], [3, 3, 2, 2] and [2, 2, 2, 2, 2].

q^11 + q^131 + q^191 + q^251 + q^311 + 2*q^371 + 2*q^431 + 3*q^491 + ...

g:=1/product((1-x^(5*j-2))*(1-x^(5*j-3)), j=1..15): gser:=series(g, x=0, 66): seq(coeff(gser, x, n), n=0..63); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2006

(PARI) {a(n) = local(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, (sqrtint(4*n + 1) - 1) \ 2, t *= x^(2*k) / (1 - x^k) * (1 + x * O(x^(n - k^2 - k))), 1), n))} /* Michael Somos Oct 15 2008 */

Cf. A003114.

Sequence in context: A058747 A050365 A029026 this_sequence A026824 A025149 A026799

Adjacent sequences: A003103 A003104 A003105 this_sequence A003107 A003108 A003109

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com), Herman P. Robinson