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Number of rooted triangular cacti with 2n+1 nodes (n triangles).
(Formerly M1448)
+10
7
1, 1, 2, 5, 13, 37, 111, 345, 1105, 3624, 12099, 41000, 140647, 487440, 1704115, 6002600, 21282235, 75890812, 272000538, 979310627, 3540297130, 12845634348, 46764904745, 170767429511, 625314778963, 2295635155206, 8447553316546, 31153444946778, 115122389065883
COMMENTS
a(n) is also the number of isomorphism classes of Fano Bott manifolds of complex dimension n (see [Cho-Lee-Masuda-Park]). - Eunjeong Lee, Jun 29 2021
REFERENCES
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 305, (4.2.34).
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.20).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n)=b(2n+1). b shifts left under transform T where Tb = EULER(E_2(b)). E_2(b) has g.f. (B(x^2)+B(x)^2)/2.
a(n) ~ c * d^n / n^(3/2), where d = 3.90053254788870206167147120260433375638561926371844809... and c = 0.4861961460367182791173441493565088408563977498871021... - Vaclav Kotesovec, Jul 01 2021
MATHEMATICA
terms = 30;
nmax = 2 terms;
A[_] = 0; Do[A[x_] = x Exp[Sum[(A[x^n]^2 + A[x^(2n)])/(2n), {n, 1, terms}]] + O[x]^nmax // Normal, {nmax}];
Number of triangular cacti with 2n+1 nodes (n triangles).
(Formerly M1152)
+10
5
1, 1, 1, 2, 4, 8, 19, 48, 126, 355, 1037, 3124, 9676, 30604, 98473, 321572, 1063146, 3552563, 11982142, 40746208, 139573646, 481232759, 1669024720, 5819537836, 20390462732, 71762924354, 253601229046, 899586777908, 3202234779826, 11435967528286, 40964243249727
REFERENCES
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 306, (4.2.35).
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.21).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: B(x)=C(x)+(C(x^3)-C(x)^3)/3.
MATHEMATICA
terms = 31;
nmax = 2 terms;
A[_] = 0;
Do[A[x_] = x Exp[Sum[(A[x^n]^2 + A[x^(2n)])/(2n), {n, 1, terms}]] + O[x]^nmax // Normal, {nmax}];
g[x_] = (A[x] /. x^k_ -> x^((k - 1)/2)) - x + 1;
Number of unrooted labeled 4-cactus graphs on 3n+1 nodes.
+10
4
1, 3, 630, 756000, 2740537800, 22317642547200, 344030189461358400, 8979238155223784448000, 366881017725878906250000000, 22141857318039212329716940800000, 1887349497873286715447530129178400000, 219275034010568207287452830493455155200000
MATHEMATICA
Table[(3 n + 1)^(n-1) (3 n)! / (2^n n!), {n, 0, 15}] (* Vincenzo Librandi, Feb 19 2020 *)
PROG
(PARI) seq(n)={my(p=serlaplace(serreverse(x*exp(-x^3/2 + O(x^(3*n+1))))/x)); vector(n+1, k, polcoef(p, 3*k-3))} \\ Andrew Howroyd, Feb 17 2020 (Magma) [(3*n+1)^(n-1)*Factorial(3*n)/(2^n*Factorial(n)): n in [0..12]]; // Vincenzo Librandi, Feb 19 2020
EXTENSIONS
a(0) changed and terms a(7) and beyond from Andrew Howroyd, Feb 17 2020
Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.
+10
4
1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 15, 1, 0, 0, 0, 0, 75, 1, 0, 0, 0, 0, 735, 280, 1, 0, 0, 0, 0, 0, 9345, 938, 1, 0, 0, 0, 0, 0, 76545, 77805, 2989, 1, 0, 0, 0, 0, 0, 0, 1865745, 536725, 9285, 1, 0, 0, 0, 0, 0, 0, 13835745, 27754650, 3334870, 28446, 1, 0
FORMULA
E.g.f.: A(x,y) = log(1 + B(x,y)) where B(x,y) is the e.g.f. of A361353. E.g.f.: A(x,y) = log(B(log(1 + x), y)/(1 + x)) where B(x,y) is the e.g.f. of A359985.
EXAMPLE
Triangle begins:
1;
0, 0;
0, 1, 0;
0, 0, 1, 0;
0, 0, 15, 1, 0;
0, 0, 0, 75, 1, 0;
0, 0, 0, 735, 280, 1, 0;
0, 0, 0, 0, 9345, 938, 1, 0;
0, 0, 0, 0, 76545, 77805, 2989, 1, 0;
...
PROG
(PARI) \\ B gives A359985 as e.g.f. B(n)= {exp(x*(1+y) + y*intformal(serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))}
T(n) = {my(v=Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))); vector(#v, n, Vecrev(v[n]/y, n))}
{ my(A=T(9)); for(i=1, #A, print(A[i])) }
E.g.f. satisfies: A(x) = exp(x^3*A(x)^3/3!).
+10
3
1, 1, 70, 28000, 33833800, 91842150400, 471920698849600, 4105733038511104000, 55918460253906250000000, 1124922893768186370457600000, 31962429471680921191680301600000, 1237813985055170041194334820761600000, 63474917512551971525535771981021376000000
FORMULA
a(n) = (3*n+1)^(n-1) * (3*n)!/(n!*(3!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^3/3!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(3*n)/(3*n)!
then a(n,m) = m*(3*n+m)^(n-1) * (3*n)!/(n!*(3!)^n).
EXAMPLE
E.g.f.: A(x) = 1 + x^3/3! + 70*x^6/6! + 28000*x^9/9! + 33833800*x^12/12! + ...
where log(A(x)) = x^3*A(x)^3/3! and
A(x)^3 = 1 + 3*x^3/3! + 270*x^6/6! + 120960*x^9/9! + 155925000*x^12/12! + ...
MATHEMATICA
Table[(3*n + 1)^(n - 1)*(3*n)!/(n!*(3!)^n), {n, 0, 30}] (* G. C. Greubel, Jul 27 2018 *)
PROG
(PARI) {a(n)=(3*n)!*polcoeff(1/x*serreverse(x*(exp(-x^3/3!+x*O(x^(3*n))))), 3*n)}
(PARI) {a(n)=(3*n+1)^(n-1)*(3*n)!/(n!*(3!)^n)}
(Magma) [(3*n+1)^(n-1)*Factorial(3*n)/(6^n*Factorial(n)): n in [0..30]]; // G. C. Greubel, Jul 27 2018 (GAP) List([0..10], n->(3*n+1)^(n-1)*Factorial(3*n)/(Factorial(n)*Factorial(3)^n)); # Muniru A Asiru, Jul 28 2018
E.g.f. satisfies: A(x) = exp(x^4*A(x)^4/4!).
+10
3
1, 1, 315, 975975, 12909521625, 495181420358625, 44035787449951171875, 7845481113748784765634375, 2526730187976408357560632640625, 1362965093449949100037985665872890625, 1160978904909328561005478318639484556796875
FORMULA
a(n) = (4*n+1)^(n-1) * (4*n)!/(n!*(4!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^4/4!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(4*n)/(4*n)!
then a(n,m) = m*(4*n+m)^(n-1) * (4*n)!/(n!*(4!)^n).
EXAMPLE
E.g.f.: A(x) = 1 + x^4/4! + 315*x^8/8! + 975975*x^12/12! + ...
where log(A(x)) = x^4*A(x)^4/4! and
A(x)^4 = 1 + 4*x^4/4! + 1680*x^8/8! + 5913600*x^12/12! + 84084000000*x^16/16! + ...
MATHEMATICA
Table[(4*n + 1)^(n - 1)*(4*n)!/(n!*(4!)^n), {n, 0, 30}] (* G. C. Greubel, Jul 27 2018 *)
PROG
(PARI) {a(n)=(4*n)!*polcoeff(1/x*serreverse(x*(exp(-x^4/4!+x*O(x^(4*n))))), 4*n)}
(PARI) {a(n)=(4*n+1)^(n-1)*(4*n)!/(n!*(4!)^n)};
(Magma) [(4*n+1)^(n-1)*Factorial(4*n)/(24^n*Factorial(n)): n in [0..30]]; // G. C. Greubel, Jul 27 2018 (GAP) List([0..10], n->(4*n+1)^(n-1)*Factorial(4*n)/(Factorial(n)*Factorial(4)^n)); # Muniru A Asiru, Jul 28 2018
E.g.f. satisfies: A(x) = exp(x^5*A(x)^5/5!).
+10
2
1, 1, 1386, 32288256, 4527372986136, 2373840824586206976, 3532226719132271834449776, 12455133709483299692008910094336, 91656142095228409912231665590704016256, 1280796898530759870923631204720457656538791936
FORMULA
a(n) = (5*n+1)^(n-1) * (5*n)! / (n!*(5!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^5/5!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(5*n)/(5*n)!
then a(n,m) = m*(5*n+m)^(n-1) * (5*n)!/(n!*(5!)^n).
EXAMPLE
E.g.f.: A(x) = 1 + x^5/5! + 1386*x^10/10! + 32288256*x^15/15! +...
where log(A(x)) = x^5*A(x)^5/5! and
A(x)^5 = 1 + 5*x^5/5! + 9450*x^10/10! + 252252000*x^15/15! + 38192529375000*x^20/20! +...
PROG
(PARI) {a(n)=(5*n)!*polcoeff(1/x*serreverse(x*(exp(-x^5/5!+x*O(x^(5*n))))), 5*n)}
(PARI) {a(n)=(5*n+1)^(n-1)*(5*n)!/(n!*(5!)^n)}
Number of rank n+1 simple connected series-parallel matroids on [2n].
+10
1
0, 1, 75, 9345, 1865745, 554479695, 231052877055, 128938132548225, 92986310399407425, 84250567868935042575, 93744545254140599193375, 125717783386887888296925825, 200041202339679732328342670625, 372688996228146502285257581079375, 803768398459351988653830600415029375
EXAMPLE
For n=2 the a(2) = 1 rank 3 simple connected series-parallel matroid on [4] is the uniform matroid of rank 3.
Triangle T read by rows: T(n, k) = k*n^(n-k-1) with 0 < k < n.
+10
0
1, 3, 2, 16, 8, 3, 125, 50, 15, 4, 1296, 432, 108, 24, 5, 16807, 4802, 1029, 196, 35, 6, 262144, 65536, 12288, 2048, 320, 48, 7, 4782969, 1062882, 177147, 26244, 3645, 486, 63, 8, 100000000, 20000000, 3000000, 400000, 50000, 6000, 700, 80, 9, 2357947691, 428717762, 58461513, 7086244, 805255, 87846, 9317, 968, 99, 10
COMMENTS
T(n, k) is the number of forests of n - k edges that connect every other labeled vertex to one of the k roots (see Section 3 in Wästlund).
REFERENCES
Alfred Rényi, Some remarks on the theory of trees. MTA Mat. Kut. Inst. Kozl. (Publ. math. Inst. Hungar. Acad. Sci) 4 (1959), 73-85.
LINKS
Arthur Cayley, A theorem on trees, Quart. J. Pure Appl. Math. 23: 376-378 (1889). Also in The collected mathematical papers of Arthur Cayley vol 13.
MATHEMATICA
Table[k*n^(n-k-1), {n, 2, 11}, {k, 1, n-1}]//Flatten
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