0,5

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.yu), 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.yu), 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

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

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].

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)=if(n<0, 0, polcoeff(sum(k=0, sqrtint(n), x^k^2/prod(i=1, k, 1-x^i, 1+x*O(x^n))), n))

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

Sequence in context: A000607 A114372 A046676 this_sequence A026823 A025148 A036821

Adjacent sequences: A003111 A003112 A003113 this_sequence A003115 A003116 A003117

easy,nonn,nice

njas, Herman P. Robinson