|
| |
|
|
A005815
|
|
Number of 4-valent labeled graphs with n nodes.
(Formerly M4991)
|
|
13
|
|
|
|
1, 0, 0, 0, 0, 1, 15, 465, 19355, 1024380, 66462606, 5188453830, 480413921130, 52113376310985, 6551246596501035, 945313907253606891, 155243722248524067795, 28797220460586826422720
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,7
|
|
|
REFERENCES
|
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 411.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f. f(x) = Sum_{n >= 0} a(n)*x^n/(n)! satisfies the differential equation 16*x^2*(x - 1)^2*(x + 2)^2*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)*(d^2/dx^2)y(x) - 4*(x^13 + 4*x^12 - 16*x^10 - 10*x^9 - 36*x^8 - 220*x^7 - 348*x^6 - 48*x^5 + 200*x^4 - 336*x^3 - 240*x^2 + 416*x - 96)*(d/dx)y(x) - x^4*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)^2*y(x) = 0.
Recurrence: a(n) = - 1/384*(( - 256*n^2 - 896*n + 1152)*a(n - 1) + (768*n^3 - 3648*n^2 + 5568*n - 2688)*a(n - 2) + ( - 192*n^4 + 3264*n^3 - 14784*n^2 + 24384*n - 12672)*a(n - 3) + (224*n^6 - 4512*n^5 + 36304*n^4 - 148160*n^3 + 320016*n^2 - 341728*n + 137856)*a(n - 5) + ( - 640*n^5 + 8800*n^4 - 46400*n^3 + 116000*n^2 - 135360*n + 57600)*a(n - 4) + ( - 24*n^10 + 1320*n^9 - 31680*n^8 + 435600*n^7 - 3786552*n^6 + 21649320*n^5 - 82006320*n^4 + 201828000*n^3 - 306085824*n^2 + 255087360*n - 87091200)*a(n - 11) + (64*n^10 - 3480*n^9 + 82692*n^8 - 1127232*n^7 + 9726024*n^6 - 55255032*n^5 + 208179908*n^4 - 510068208*n^3 + 770738352*n^2 - 640484928*n + 218211840)*a(n - 9) + (16*n^11 - 992*n^10 + 27256*n^9 - 437160*n^8 + 4536288*n^7 - 31876656*n^6 + 154182488*n^5 - 510784360*n^4 + 1128552896*n^3 - 1570313952*n^2 + 1223830656*n - 397716480)*a(n - 10) + ( - 128*n^8 + 5488*n^7 - 94576*n^6 + 864976*n^5 - 4606672*n^4 + 14604352*n^3 - 26753984*n^2 + 25611264*n - 9630720)*a(n - 7) + (16*n^9 - 576*n^8 + 8704*n^7 - 71680*n^6 + 348880*n^5 - 1013824*n^4 + 1673376*n^3 - 1333120*n^2 + 226944*n + 161280)*a(n - 8) + (128*n^7 - 2192*n^6 + 12048*n^5 - 8240*n^4 - 151248*n^3 + 565312*n^2 - 765248*n + 349440)*a(n - 6) + ( - 4*n^13 + 364*n^12 - 14924*n^11 + 364364*n^10 - 5897892*n^9 + 66678612*n^8 - 540145892*n^7 + 3163772612*n^6 - 13344475144*n^5 + 39830815024*n^4 - 81255012384*n^3 + 106386868224*n^2 - 79211036160*n + 24908083200)*a(n - 14) + ( - 4*n^13 + 360*n^12 - 14612*n^11 + 353496*n^10 - 5674812*n^9 + 63680760*n^8 - 512439356*n^7 + 2983811688*n^6 - 12520194544*n^5 + 37201987680*n^4 - 75598952832*n^3 + 98660630016*n^2 - 73265264640*n + 22992076800)*a(n - 13) + ( - 16*n^12 + 1244*n^11 - 43208*n^10 + 884620*n^9 - 11860728*n^8 + 109396452*n^7 - 709293464*n^6 + 3243764260*n^5 - 10331326456*n^4 + 22203205904*n^3 - 30301280928*n^2 + 23300910720*n - 7504358400)*a(n - 12) + ( - n^14 + 105*n^13 - 5005*n^12 + 143325*n^11 - 2749747*n^10 + 37312275*n^9 - 368411615*n^8 + 2681453775*n^7 - 14409322928*n^6 + 56663366760*n^5 - 159721605680*n^4 + 310989260400*n^3 - 392156797824*n^2 + 283465647360*n - 87178291200)*a(n - 15)). (End)
a(n) = Sum_{d=0..floor(n/2), c=0..floor(n/2-d), b=0..(n-2c-2d), f=0..(n-2c-2d-b), k=0..min(n-b-2c-2d-f, 2n-2f-2b-3c-4d), j=0..floor(k/2+f)} ((-1)^(k+2f-j+d)*n!*(k+2f)!(2(2n-k-2f-2b-3c-4d))!) / (2^(5n-2k-2f-3b-8c-7d) * 3^(n-b-c-2d-k-f)*(2n-k-2f-2b-3c-4d)!*(k+2f-2j)!*j!*b!*c!*d!*k!*f!*(n-b-2c-2d-k-f)!). - Shanzhen Gao, Jun 05 2009
E.g.f.: (1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4],[],-12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16)). - Mark van Hoeij, Nov 07 2011
a(n) ~ n^(2*n) * 2^(n+1/2) / (3^n * exp(2*n+15/4)). - Vaclav Kotesovec, Mar 11 2014
|
|
|
MAPLE
|
egf := (1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4], [], -12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16));
ser := convert(series(egf, x=0, 40), polynom):
seq(coeff(ser, x, i)*i!, i=0..degree(ser)); # Mark van Hoeij, Nov 07 2011
|
|
|
MATHEMATICA
|
max = 17; f[x_] := HypergeometricPFQ[{1/4, 3/4}, {}, -12*x*(x + 2)*(x - 1)/(x^3 + 2*x^2 - 6*x - 6)^2]*Exp[-x*(x^2 - 6)/(8*x + 16)]/(1 + x - x^2/3 - x^3/6)^ (1/2); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Jun 19 2012, from e.g.f. *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
