A052254 - OEIS

%I #20 Sep 08 2022 08:44:59

%S 1,17,108,444,1410,3762,8844,18876,37323,69355,122408,206856,336804,

%T 531012,813960,1217064,1780053,2552517,3595636,4984100,6808230,

%U 9176310,12217140,16082820,20951775

%N Partial sums of A050406.

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

%D Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

%H G. C. Greubel, <a href="/A052254/b052254.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 a(n) = (10*n + 7)*binomial(n+6, 6)/7.

%F G.f.: (1+9*x)/(1-x)^8.

%F From _G. C. Greubel_, Jan 19 2020: (Start)

%F a(n) = 10*binomial(n+7, 7) - 9*binomial(n+6, 6).

%F E.g.f.: (7! + 80640*x + 189000*x^2 + 142800*x^3 + 45150*x^4 + 6552*x^5 + 427*x^6 + 10*x^7)*exp(x)/7!. (End)

%F 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). - _Wesley Ivan Hurt_, Nov 28 2021

%p seq( (10*n+7)*binomial(n+6, 6)/7, n=0..30); # _G. C. Greubel_, Jan 19 2020

%t Table[10*Binomial[n+7,7] -9*Binomial[n+6,6], {n,0,30}] (* _G. C. Greubel_, Jan 19 2020 *)

%t Rest[Nest[Accumulate[#]&,Table[n(n+1)(10n-7)/6,{n,0,50}],4]] (* _Harvey P. Dale_, Aug 03 2020 *)

%o (PARI) vector(31, n, (10*n-3)*binomial(n+5, 6)/7) \\ _G. C. Greubel_, Jan 19 2020

%o (Magma) [(10*n+7)*Binomial(n+6, 6)/7: n in [0..30]]; // _G. C. Greubel_, Jan 19 2020

%o (Sage) [(10*n+7)*binomial(n+6, 6)/7 for n in (0..30)] # _G. C. Greubel_, Jan 19 2020

%o (GAP) List([0..30], n-> (10*n+7)*Binomial(n+6, 6)/7 ); # _G. C. Greubel_, Jan 19 2020

%Y Cf. A050406.

%Y Cf. A093645 ((10, 1) Pascal, column m=7).

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Feb 03 2000