0,3

a(n) counts stacks of integer-length boards of total length n where no board overhangs the board underneath.

A006330(n)+a(n)=A000712(n). - Michael Somos, Jul 22 2003

Number of graphical partitions on 2n nodes that contain a 1. E.g. a(3)=4, and so there are 4 graphical partitions of 6 that contain a 1, namely (111111), (21111), (2211) and (3111). Only (222) fails. - Jon Perry (perry(AT)globalnet.co.uk), Jul 25 2003

F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs. Proc. Cambridge Philos. Soc. 47, (1951), 679-686.

E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.

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

H. Bottomley, Illustration of initial terms

a(n) = Sum(1 <= k <= n, f(k, n-k)), where f(n, k) (=A054250) = 1 if k = 0; Sum(1 <= j <= min(n, k); (n-j+1) f(j, k-j)) if k > 0.

a(n)=sum_k[A059623(n, k)] for n>0 - Henry Bottomley (se16(AT)btinternet.com), Feb 01 2001

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

For a(4)=8 we have the following stacks:

x

x x. .x

x x. .x x.. .x. ..x xx

x xx xx xxx xxx xxx xx xxxx

(PARI) a(n)=if(n<1, 0, polcoeff(sum(k=1, (sqrt(1+8*n)-1)\2, -(-1)^k*x^((k+k^2)/2))/eta(x+x*O(x^n))^2, n))

Cf. A054250, A059618, A059623, A001522, A001524.

Cf. A000569. Bisections give A100505, A100506.

Adjacent sequences: A001520 A001521 A001522 this_sequence A001524 A001525 A001526

Sequence in context: A003241 A125513 A054174 this_sequence A000126 A098057 A074029

nonn,nice,easy

njas

Formula and more terms from David W. Wilson (davidwwilson(AT)comcast.net) May 05 2000.