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

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

Table of n, a(n) for n=1..36.

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: A338249 A214271 A262007 * A015152 A021461 A075733

Adjacent sequences: A005486 A005487 A005488 * A005490 A005491 A005492

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