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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

References

• A. 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).

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.

Programs

GAP

List([0..30], n-> (6*n+7)*Binomial(n+6,6)/7); # G. C. Greubel, Aug 28 2019

Magma

[(6*n+7)*Binomial(n+6,6)/7: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

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 *)

PARI

a(n)=(6*n/7+1)*binomial(n+6,6) \\ Charles R Greathouse IV, Oct 07 2015

Sage

[(6*n+7)*binomial(n+6,6)/7 for n in (0..30)] # G. C. Greubel, Aug 28 2019

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

Extensions

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