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

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).

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

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46

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.

Sequence in context: A003241 A125513 A054174 this_sequence A000126 A143281 A098057

Adjacent sequences: A001520 A001521 A001522 this_sequence A001524 A001525 A001526

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

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