Displaying 1-10 of 18 results found.
DIK(b)-DIK[ 2 ](b)-b where b is A035082.
+20
5
0, 0, 0, 1, 1, 2, 3, 7, 14, 33, 74, 180, 438, 1090, 2741, 6994, 17966, 46565, 121440, 318597, 839953, 2224486, 5914248, 15780662, 42241422, 113402369, 305254039, 823690961, 2227640597, 6037142355, 16392945284, 44592703836
PROG
(PARI)
BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
DIK(p, n)={(sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d))) + ((1+p)^2/(1-subst(p, x, x^2))-1)/2)/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); Vec(DIK(p, n) - p - (p^2 + subst(p, x, x^2))/2, -(n+1))} \\ Andrew Howroyd, Aug 31 2018
0, 0, 1, 1, 2, 4, 9, 20, 47, 112, 273, 676, 1694, 4296, 10991, 28350, 73614, 192327, 505093, 1332801, 3531598, 9393501, 25070735, 67121670, 180216260, 485133376, 1309101329, 3540394176, 9594562328, 26051397890, 70861839620
PROG
(PARI)
BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); concat([0], Vec(BIK(p)-1)-Vec(p))} \\ Andrew Howroyd, Aug 30 2018
Number of mixed Husimi trees with n nodes; or polygonal cacti with bridges.
(Formerly M1191 N0461)
+10
14
1, 1, 1, 2, 4, 9, 23, 63, 188, 596, 1979, 6804, 24118, 87379, 322652, 1209808, 4596158, 17657037, 68497898, 268006183, 1056597059, 4193905901, 16748682185, 67258011248, 271452424286, 1100632738565, 4481533246014
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Number of mixed Husimi trees with n nodes; or labeled polygonal cacti with bridges.
(Formerly M3639 N1480)
+10
13
1, 1, 1, 4, 31, 362, 5676, 111982, 2666392, 74433564, 2384579440, 86248530296, 3476794472064, 154579941792256, 7514932528712896, 396595845237540600, 22581060079942183936, 1379771773100463174608, 90059660791562688208128, 6253914166368448348512064
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
A:= proc(n) option remember; if n<=0 then x else convert(series(x* exp((2*A(n-1) -A(n-1)^2)/ (2-2*A(n-1))), x=0, n+2), polynom) fi end: a:= n-> if n=0 then 1 else coeff(series(A(n-1), x=0, n+1), x, n)*(n-1)! fi: seq(a(n), n=0..30); # Alois P. Heinz, Aug 20 2008
MATHEMATICA
A[n_] := A[n] = If[n <= 0, x, Normal[Series[x*Exp[(2*A[n-1]-A[n-1]^2)/ (2-2*A[n-1])], {x, 0, n+2}]]]; a[n_] := If[n == 0, 1, Coefficient [Series[A[n-1], {x, 0, n+1}], x, n]*(n-1)!]; Table [a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Number of mixed Husimi trees with n nodes; or rooted polygonal cacti with bridges.
(Formerly M2754 N1107)
+10
12
0, 1, 1, 3, 8, 26, 84, 297, 1066, 3976, 15093, 58426, 229189, 910127, 3649165, 14756491, 60103220, 246357081, 1015406251, 4205873378, 17497745509, 73084575666, 306352303774, 1288328048865, 5433980577776, 22982025183983
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
Shifts left under transform T where Ta = EULER(BIK(a)). [See Transforms links.] - Christian G. Bower, Nov 15 1998
PROG
(PARI)
BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[0]); for(n=1, n, v=concat([0, 1], EulerT(Vec(BIK(Ser(v))-1)))); v} \\ Andrew Howroyd, Aug 30 2018
"DIK" (bracelet, indistinct, unlabeled) transform of A000237.
+10
12
1, 1, 2, 5, 14, 43, 143, 496, 1794, 6667, 25345, 98032, 384713, 1527480, 6125327, 24770186, 100897860, 413595904, 1704840125, 7062024986, 29382224119, 122731488819, 514491387498, 2163757816681, 9126920239124, 38602653740841
Number of increasing asymmetric rooted polygonal cacti with bridges (mixed Husimi trees).
+10
11
1, 1, 1, 7, 39, 409, 4687, 62822, 945250, 15999616, 300150210, 6198330586, 139779046596, 3420083177362, 90241503643208, 2554721759776914, 77240614583288344, 2484170781778551036
FORMULA
Shifts left under transform T where Ta = EGJ(BHJ(a)).
Number of polygonal cacti (Husimi graphs) with n nodes.
+10
7
1, 1, 0, 1, 1, 2, 2, 5, 7, 16, 28, 63, 131, 301, 673, 1600, 3773, 9158, 22319, 55255, 137563, 345930, 874736, 2227371, 5700069, 14664077, 37888336, 98310195, 256037795, 669184336, 1754609183, 4614527680
REFERENCES
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.
PROG
(PARI)
BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
DIK(p, n)={(sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d))) + ((1+p)^2/(1-subst(p, x, x^2))-1)/2)/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); Vec(1 + DIK(p, n) - (p^2 + subst(p, x, x^2))/2 - p*(BIK(p)-1-p))} \\ Andrew Howroyd, Aug 31 2018
Number of labeled polygonal cacti (Husimi graphs) with n nodes.
+10
4
1, 1, 0, 1, 3, 27, 240, 2985, 42840, 731745, 14243040, 313570845, 7683984000, 207685374435, 6135743053440, 196754537704725, 6805907485977600, 252620143716765825, 10015402456976716800, 422410127508300756825, 18884777200534941696000
COMMENTS
A Husimi tree is a connected graph in which no line lies on more than one cycle [Harary, 1953]. - Jonathan Vos Post, Mar 12 2010
REFERENCES
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134-141.
F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.
F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39. pp. 315-322, 1953.
F. Harary, G. Uhlenbeck (1953), "On the number of Husimi trees, I", Proceedings of the National Academy of Sciences 39: 315-322. - From Jonathan Vos Post, Mar 12 2010
MATHEMATICA
max = 20; s = 1+InverseSeries[Series[E^(x^2/(2*(x-1)))*x, {x, 0, max}], x]; a[n_] := SeriesCoefficient[s, n]*(n-1)!; a[0]=1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 27 2016, after Vaclav Kotesovec at A035087 *)
Number of asymmetric rooted polygonal cacti with bridges (mixed Husimi trees).
+10
4
0, 1, 1, 1, 3, 7, 22, 67, 215, 692, 2283, 7599, 25631, 87211, 299386, 1035059, 3602083, 12606318, 44344764, 156698081, 555989604, 1980050697, 7075365521, 25360341963, 91155701023, 328500571740, 1186656421109, 4296084607302
FORMULA
Shifts left under transform T where Ta = WEIGH(BHK(a)).
PROG
(PARI)
BHK(p)={p + (1/(1-p) - (1+p)/subst(1-p, x, x^2))/2}
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(v=[0]); for(n=1, n, v=concat([0, 1], WeighT(Vec(BHK(Ser(v)))))); v} \\ Andrew Howroyd, Aug 30 2018
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