A034265 - OEIS

%I #30 Sep 08 2022 08:44:51

%S 1,13,76,300,930,2442,5676,12012,23595,43615,76648,129064,209508,

%T 329460,503880,751944,1097877,1571889,2211220,3061300,4177030,5624190,

%U 7480980,9839700,12808575,16513731,21101328,26739856,33622600,41970280

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

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

%H Vincenzo Librandi, <a href="/A034265/b034265.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

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

%F 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

%p seq((6*n+7)*binomial(n+6,6)/7, n=0..30); # _G. C. Greubel_, Aug 28 2019

%t 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 *)

%t CoefficientList[Series[(1+5x)/(1-x)^8, {x,0,40}], x] (* _Vincenzo Librandi_, Jul 30 2014 *)

%o (Magma) [(6*n+7)*Binomial(n+6,6)/7: n in [0..40]]; // _Vincenzo Librandi_, Jul 30 2014

%o (PARI) a(n)=(6*n/7+1)*binomial(n+6,6) \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Sage) [(6*n+7)*binomial(n+6,6)/7 for n in (0..30)] # _G. C. Greubel_, Aug 28 2019

%o (GAP) List([0..30], n-> (6*n+7)*Binomial(n+6,6)/7); # _G. C. Greubel_, Aug 28 2019

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

%Y Partial sums of A027810.

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

%Y Cf. similar sequences listed in A254142.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_

%E Corrected and extended by _N. J. A. Sloane_, Apr 21 2000