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).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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? - N. J. A. Sloane (njas(AT)research.att.com)]

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.

Sequence in context: A119776 A087265 A079307 this_sequence A087259 A083251 A075690

Adjacent sequences: A005811 A005812 A005813 this_sequence A005815 A005816 A005817

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

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

Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 19 2007