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A005489
Number of nonzero coefficients of order n in Baker-Campbell-Hausdorff expansion.
(Formerly M0181)
1
1, 1, 2, 1, 8, 7, 32, 31, 96, 97, 512, 511, 2048, 2047, 7396, 7531, 32768, 32767, 131072, 131071, 508436, 512245, 2097152, 2097151, 8202208, 8207797, 33256980, 33335611, 134217728, 134217727, 536870912, 536870911, 2142108916, 2143603741, 8589928768, 8589921949
OFFSET
1,3
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
J. A. Oteo, The Baker-Campbell-Hausdorff formula and nested commutator relations, J. Math. Phys., 32 (1991), 419-421.
J. A. Oteo, The Baker-Campbell-Hausdorff formula and nested commutator identities, J. Math. Phys., 32.2 (1991), 419, 421. (Annotated scanned copy)
MATHEMATICA
g[1] = 1;
g[s_] := g[s] = Expand[(2 t - 1) g[s - 1] + t (t - 1) D[g[s - 1], t]];
cx[ss_] := Module[{m, mp, mpp, \[Gamma]},
m = Length[ss] + 1;
mp = Floor[m/2];
mpp = Floor[(m - 1)/2];
\[Gamma] = CoefficientList[Product[g[s], {s, ss}], t];
(-1)^mpp mpp! / Product[s!, {s, ss}] Sum[\[Gamma][[k]] (mp + k - 1)!/(m + k - 1)!, {k, Total[ss] - m + 2}]
];
cxs[n_] := Select[Table[{cx[ss], Length@Permutations@ss}, {ss, IntegerPartitions[n - 1]}], First@# != 0 &];
a[n_] := Total[Last /@ cxs[n]];
Table[a[n], {n, 10}]
(* Andrey Zabolotskiy, Dec 27 2018 *)
CROSSREFS
Sequence in context: A214271 A372474 A262007 * A015152 A021461 A075733
KEYWORD
nonn,nice
AUTHOR
David J. Thompson
EXTENSIONS
a(11)-a(30) from Andrey Zabolotskiy, Dec 27 2018
a(31)-a(36) from Sean A. Irvine, Feb 26 2021
STATUS
approved