0,5

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

Same as number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 2.

As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001

The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))).

Coefficients in expansion of permanent of infinite tridiagonal matrix:

1 1 0 0 0 0 0 0 ...

x 1 1 0 0 0 0 0 ...

0 x^2 1 1 0 0 0 ...

0 0 x^3 1 1 0 0 ...

0 0 0 x^4 1 1 0 ...

................... - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 17 2004

Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 17 2004

Also number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006

Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3, 3, 3], [2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 16 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. 109, 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(e), 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. 107.

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

A. V. Sills, Finite Rogers-Ramanujan type identities. Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp.

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

G. E. Andrews, Three aspects of partitions

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.

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

G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 06 2004

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

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

a(9)=5 because we have [9], [6, 1, 1, 1], [4, 4, 1], [4, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

1/q + q^59 + q^119 + q^179 + 2*q^239 + 2*q^299 + 3*q^359 + 3*q^419 + ...

g:=sum(x^(k^2)/product(1-x^j, j=1..k), k=0..10): gser:=series(g, x=0, 65): seq(coeff(gser, x, n), n=0..60); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006

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

Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900.

Sequence in context: A000607 A114372 A046676 this_sequence A026823 A025148 A096749

Adjacent sequences: A003111 A003112 A003113 this_sequence A003115 A003116 A003117

easy,nonn,nice

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