|
%I #15 Sep 08 2022 08:45:03
%S 5,56,263,815,1982,4115,7646,13088,21035,32162,47225,67061,92588,
%T 124805,164792,213710,272801,343388,426875,524747,638570,769991,
%U 920738,1092620,1287527,1507430,1754381,2030513,2338040,2679257,3056540
%N Fifth column of A046741.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).
%H G. C. Greubel, <a href="/A062125/b062125.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F G.f.: (5 + 33*x^2 + 10*x^3 + 31*x + 2*x^4)/(1-x)^5. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y)-(1+y)*x).
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), where a(0)=5, a(1)=56, a(2)=263, a(3)=815, a(4)=1982. - _Harvey P. Dale_, Dec 21 2011
%F From _G. C. Greubel_, Jan 31 2019: (Start)
%F a(n) = (40 + 126*n + 165*n^2 + 90*n^3 + 27*n^4)/8.
%F E.g.f.: (40 + 408*x + 624*x^2 + 252*x^3 + 27*x^4)*exp(x)/8. (End)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {5, 56, 263, 815, 1982}, 31] (* or *) CoefficientList[Series[(5+33x^2+10x^3+31x+2x^4)/(1-x)^5,{x,0,30}],x] (* _Harvey P. Dale_, Dec 21 2011 *)
%t Table[(40+126*n+165*n^2+90*n^3+27*n^4)/8, {n,0,40}] (* _G. C. Greubel_, Jan 31 2019 *)
%o (PARI) vector(40, n, n--; (40+126*n+165*n^2+90*n^3+27*n^4)/8) \\ _G. C. Greubel_, Jan 31 2019
%o (Magma) [(40+126*n+165*n^2+90*n^3+27*n^4)/8: n in [0..40]]; // _G. C. Greubel_, Jan 31 2019
%o (Sage) [(40+126*n+165*n^2+90*n^3+27*n^4)/8 for n in range(40)] # _G. C. Greubel_, Jan 31 2019
%o (GAP) List([0..40], n -> (40+126*n+165*n^2+90*n^3+27*n^4)/8); # _G. C. Greubel_, Jan 31 2019
%Y Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.
%K easy,nonn
%O 0,1
%A _Vladeta Jovovic_, Jun 04 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
|
