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 A005489 Number of nonzero coefficients of order n in Baker-Campbell-Hausdorff expansion. (Formerly M0181) 1

%I M0181

%S 1,1,2,1,8,7,32,31,96,97,512,511,2048,2047,7396,7531,32768,32767,

%T 131072,131071,508436,512245,2097152,2097151,8202208,8207797,33256980,

%U 33335611,134217728,134217727,536870912,536870911,2142108916,2143603741,8589928768,8589921949

%N Number of nonzero coefficients of order n in Baker-Campbell-Hausdorff expansion.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H J. A. Oteo, <a href="http://dx.doi.org/10.1063/1.529428">The Baker-Campbell-Hausdorff formula and nested commutator relations</a>, J. Math. Phys., 32 (1991), 419-421.

%H J. A. Oteo, <a href="/A005489/a005489.pdf">The Baker-Campbell-Hausdorff formula and nested commutator identities</a>, J. Math. Phys., 32.2 (1991), 419, 421. (Annotated scanned copy)

%t g[1] = 1;

%t g[s_] := g[s] = Expand[(2 t - 1) g[s - 1] + t (t - 1) D[g[s - 1], t]];

%t cx[ss_] := Module[{m, mp, mpp, \[Gamma]},

%t m = Length[ss] + 1;

%t mp = Floor[m/2];

%t mpp = Floor[(m - 1)/2];

%t \[Gamma] = CoefficientList[Product[g[s], {s, ss}], t];

%t (-1)^mpp mpp! / Product[s!, {s, ss}] Sum[\[Gamma][[k]] (mp + k - 1)!/(m + k - 1)!, {k, Total[ss] - m + 2}]

%t ];

%t cxs[n_] := Select[Table[{cx[ss], Length@Permutations@ss}, {ss, IntegerPartitions[n - 1]}], First@# != 0 &];

%t a[n_] := Total[Last /@ cxs[n]];

%t Table[a[n], {n, 10}]

%t (* _Andrey Zabolotskiy_, Dec 27 2018 *)

%K nonn,nice

%O 1,3

%A David J. Thompson

%E a(11)-a(30) from _Andrey Zabolotskiy_, Dec 27 2018

%E a(31)-a(36) from _Sean A. Irvine_, Feb 26 2021

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Last modified June 19 18:24 EDT 2021. Contains 345144 sequences. (Running on oeis4.)