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Number of n-node connected Stanley graphs.
5

%I #18 May 24 2020 06:32:36

%S 0,1,1,2,8,52,502,6824,127166,3205924,108975934,5006366048,

%T 312601245662,26708244267148,3142852107059758,512229404374936616,

%U 116165284523764481294,36791597841822774872116,16320947226945992981680606,10163558457757761048966068912

%N Number of n-node connected Stanley graphs.

%C For precise definition see Knuth (1997).

%H Alois P. Heinz, <a href="/A323843/b323843.txt">Table of n, a(n) for n = 0..155</a>

%H D. E. Knuth, <a href="/A323841/a323841.pdf">Letter to Daniel Ullman and others</a>, Apr 29 1997. [Annotated scanned copy, with permission]

%p b:= proc(n) option remember; add(mul(

%p (2^(i+k)-1)/(2^i-1), i=1..n-k), k=0..n)

%p end:

%p p:= proc(n) option remember;

%p add(b(n-j)*binomial(n, j)*(-1)^j, j=0..n)

%p end:

%p a:= proc(n) option remember; `if`(n=0, 0, p(n)-add(

%p binomial(n, j)*p(n-j)*a(j)*j, j=1..n-1)/n)

%p end:

%p seq(a(n), n=0..21); # _Alois P. Heinz_, Sep 24 2019

%t b[n_] := b[n] = Sum[Product[(2^(i+k) - 1)/(2^i - 1), {i, n-k}], {k, 0, n}];

%t p[n_] := p[n] = Sum[b[n-j] Binomial[n, j] (-1)^j, {j, 0, n}];

%t a[n_] := a[n] = If[n == 0, 0, p[n] - Sum[Binomial[n, j] p[n-j] a[j] j, {j, n-1}]/n];

%t a /@ Range[0, 21] (* _Jean-François Alcover_, May 24 2020, after _Alois P. Heinz_ *)

%Y Cf. A135922, A323841, A323842.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Feb 04 2019

%E More terms from _Alois P. Heinz_, Sep 24 2019