A027658 a(n) = binomial(n+2, 2) + binomial(n+4, 5).

1, 4, 12, 31, 71, 147, 280, 498, 837, 1342, 2068, 3081, 4459, 6293, 8688, 11764, 15657, 20520, 26524, 33859, 42735, 53383, 66056, 81030, 98605, 119106, 142884, 170317, 201811, 237801, 278752, 325160, 377553, 436492, 502572, 576423, 658711, 750139, 851448

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000217, A000389.

Programs

Magma

[(60 +12*n +7*n^2 +n^3)*Binomial(n+2,2)/60: n in [0..60]]; // G. C. Greubel, Aug 01 2022

Mathematica

CoefficientList[Series[(1-2*x+3*x^2-x^3)/(1-x)^6, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
Sum[Binomial[2*j +Range[0, 60], 3*j-1], {j,2}] (* G. C. Greubel, Aug 01 2022 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,4,12,31,71,147},40] (* Harvey P. Dale, Feb 27 2023 *)

SageMath

[(60 +12*n +7*n^2 +n^3)*binomial(n+2,2)/60 for n in (0..60)] # G. C. Greubel, Aug 01 2022

Formula

G.f.: (1 - 2*x + 3*x^2 - x^3)/(1-x)^6. - Colin Barker, Apr 15 2013

From R. J. Mathar, Sep 29 2020: (Start)

a(n) = A000217(n+1) + A000389(n+4)

a(n) = (n+1)*(n+2)*(60 +12*n +7*n^2 +n^3)/120. (End)

From G. C. Greubel, Aug 01 2022: (Start)

a(n) = Sum_{j=1..2} binomial(n+2*j, 3*j-1).

E.g.f.: (1/120)*(120 + 360*x + 300*x^2 + 120*x^3 + 20*x^4 + x^5)*exp(x). (End)