A062125 Fifth column of A046741.

5, 56, 263, 815, 1982, 4115, 7646, 13088, 21035, 32162, 47225, 67061, 92588, 124805, 164792, 213710, 272801, 343388, 426875, 524747, 638570, 769991, 920738, 1092620, 1287527, 1507430, 1754381, 2030513, 2338040, 2679257, 3056540

Offset: 0

Author

Vladeta Jovovic, Jun 04 2001

Keywords

• easy
• nonn

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Crossrefs

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

Programs

GAP

List([0..40], n -> (40+126*n+165*n^2+90*n^3+27*n^4)/8); # G. C. Greubel, Jan 31 2019

Magma

[(40+126*n+165*n^2+90*n^3+27*n^4)/8: n in [0..40]]; // G. C. Greubel, Jan 31 2019

Mathematica

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 *)
Table[(40+126*n+165*n^2+90*n^3+27*n^4)/8, {n,0,40}] (* G. C. Greubel, Jan 31 2019 *)

PARI

vector(40, n, n--; (40+126*n+165*n^2+90*n^3+27*n^4)/8) \\ G. C. Greubel, Jan 31 2019

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

Formula

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).

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

From G. C. Greubel, Jan 31 2019: (Start)

a(n) = (40 + 126*n + 165*n^2 + 90*n^3 + 27*n^4)/8.

E.g.f.: (40 + 408*x + 624*x^2 + 252*x^3 + 27*x^4)*exp(x)/8. (End)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001