0,2

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Goulden, I. P.; Jackson, D. M.; Reilly, J. W.; The Hammond series of a symmetric function and its application to $P$-recursiveness. SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 175, (7.5.12).

E.g.f. f(x) = Sum_{n >= 0} a(2 * n) * x^n/(2 * n)! satisfies differential equation 6 * x^2 * (x^2 - 2 * x - 2) * diff(f(x), x, x) - (x^5 - 6 * x^4 + 6 * x^3 + 24 * x^2 + 16 * x - 8) * diff(f(x), x) + 1/6 * (x^5 - 10 * x^4 + 24 * x^3 - 4 * x^2 - 44 * x - 48) * f(x) = 0.

Recurrence: a(2 * n) = (2 * n)!/n! * v(n) where 48 * v(n) + ( - 72 * n^2 + 120 * n - 96) * v(n - 1) + ( - 72 * n^3 + 288 * n^2 - 404 * n + 188) * v(n - 2) + (36 * n^4 - 396 * n^3 + 1472 * n^2 - 2184 * n + 1072) * v(n - 3) + (36 * n^4 - 336 * n^3 + 1116 * n^2 - 1536 * n + 720) * v(n - 4) + ( - 6 * n^5 + 80 * n^4 - 410 * n^3 + 1000 * n^2 - 1144 * n + 480) * v(n - 5) + (n^5 - 15 * n^4 + 85 * n^3 - 225 * n^2 + 274 * n - 120) * v(n - 6) = 0.

Linear recurrence satisfied by a(n): {a(0) = 1, a(1) = 2, a(2) = 47, a(3) = 4720, a(4) = 1256395, a(5) = 699971370 and

(4989600 + 5718768*n^7 + 1045440*n^8 + 123200*n^9 + 8448*n^10 + 256*n^11 + 30135960*n + 75458988*n^2 + 105258076*n^3 + 91991460*n^4 + 53358140*n^5 + 21100464*n^6)*a(n) + ( - 39916800 - 1756320*n^7 - 198720*n^8 - 13120*n^9 - 384*n^10 - 136306080*n - 205327944*n^2 - 179845580*n^3 - 101513280*n^4 - 38608500*n^5 - 10026072*n^6)*a(n + 1) + (19958400 + 17664*n^7 + 576*n^8 + 44868240*n + 43664892*n^2 + 24024336*n^3 + 8173284*n^4 + 1760640*n^5 + 234528*n^6)*a(n + 2) + (720720 + 144*n^7 + 1819364*n + 1758924*n^2 + 883226*n^3 + 254070*n^4 + 42356*n^5 + 3816*n^6)*a(n + 3) + ( - 183645 - 191119*n - 79608*n^2 - 16586*n^3 - 1728*n^4 - 72*n^5)*a(n + 4) + ( - 2706 - 1515*n - 285*n^2 - 18*n^3)*a(n + 5) + 3*a(n + 6)}. - Marni Mishna (marni.mishna(AT)inria.fr), Jun 17 2005

Linear differential equation satisfied by F(t)=Sum a(n) t^n/(2n)!: {F(0) = 1, - 3*t*(10*t^2 + 9*t^6 + 18*t^4 - 8 + t^10 - 6*t^8)*( - 2 - 2*t^2 + t^4)*(diff(F(t), t)) + 9*t^4*( - 2 - 2*t^2 + t^4)^2*(diff(F(t), `$`(t, 2))) + t^2*( - 2 - 2*t^2 + t^4)*(24*t^6 - 10*t^8 - 4*t^4 - 44*t^2 + t^10 - 48)*F(t)}. - Marni Mishna (marni.mishna(AT)inria.fr), Jun 17 2005 [Probably this defines A005814? - njas]

Equation (7.5.13) in Harary and Palmer gives asymptotic formula.

a(1)=2: {(1,1), (1,2), (2,2)}, {(1,2), (1,2), (1,2)}

Cf. A002829, A002135.

Adjacent sequences: A005811 A005812 A005813 this_sequence A005815 A005816 A005817

Sequence in context: A119776 A087265 A079307 this_sequence A087259 A083251 A075690

nonn,easy,nice

njas, Simon Plouffe (plouffe(AT)math.uqam.ca)

More terms and formulae from Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 25 2001

Edited by njas, Apr 19 2007