A034265 a(n) = binomial(n+6,6)*(6*n+7)/7.

1, 13, 76, 300, 930, 2442, 5676, 12012, 23595, 43615, 76648, 129064, 209508, 329460, 503880, 751944, 1097877, 1571889, 2211220, 3061300, 4177030, 5624190, 7480980, 9839700, 12808575, 16513731, 21101328, 26739856, 33622600, 41970280, 52033872, 64097616, 78482217

Offset

0,2

References

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Formula

G.f.: (1+5*x)/(1-x)^8.

a(0)=1, a(1)=13, a(2)=76, a(3)=300, a(4)=930, a(5)=2442, a(6)=5676, a(7)=12012, a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Jul 29 2014

From Amiram Eldar, Sep 27 2025: (Start)

Sum_{n>=0} 1/a(n) = 226224033/623645 - 3919104*sqrt(3)*Pi/124729 - 11757312*log(3)/124729 - 15676416*log(2)/124729.

Sum_{n>=0} (-1)^n/a(n) = 7838208*Pi/124729 - 9163392*log(2)/124729 + 7838208*sqrt(3)*log(2+sqrt(3))/124729 - 180178348/623645. (End)

From Enrique Navarrete, Feb 02 2026: (Start)

a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(6*n+7)/5040.

E.g.f.: (1/5040)*(6*x^7 + 259*x^6 + 4032*x^5 + 28350*x^4 + 92400*x^3 + 128520*x^2 + 60480*x + 5040)*exp(x). (End)

Maple

seq((6*n+7)*binomial(n+6, 6)/7, n=0..30); # G. C. Greubel, Aug 28 2019

Mathematica

Accumulate[Table[(n+1)Binomial[n+5, 5], {n, 0, 30}]] (* or *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {1, 13, 76, 300, 930, 2442, 5676, 12012}, 30] (* Harvey P. Dale, Jul 29 2014 *)
CoefficientList[Series[(1+5x)/(1-x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

Prog

(Magma) [(6*n+7)*Binomial(n+6, 6)/7: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014
(PARI) a(n)=(6*n/7+1)*binomial(n+6, 6) \\ Charles R Greathouse IV, Oct 07 2015
(SageMath) [(6*n+7)*binomial(n+6, 6)/7 for n in (0..30)] # G. C. Greubel, Aug 28 2019
(GAP) List([0..30], n-> (6*n+7)*Binomial(n+6, 6)/7); # G. C. Greubel, Aug 28 2019

Crossrefs

a(n) = f(n, 5) where f is given in A034261.

Partial sums of A027810.

Cf. A093563 ((6, 1) Pascal, column m=7).

Cf. similar sequences listed in A254142.

Sequence in context: A005340 A114244 A050485 * A282643 A269085 A182077

Adjacent sequences: A034262 A034263 A034264 * A034266 A034267 A034268

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Corrected and extended by N. J. A. Sloane, Apr 21 2000

Status

approved