A052254 Partial sums of A050406.

1, 17, 108, 444, 1410, 3762, 8844, 18876, 37323, 69355, 122408, 206856, 336804, 531012, 813960, 1217064, 1780053, 2552517, 3595636, 4984100, 6808230, 9176310, 12217140, 16082820, 20951775

Offset: 0

Author

Barry E. Williams, Feb 03 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Links

• G. C. Greubel, 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

Cf. A050406.

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

Programs

GAP

List([0..30], n-> (10*n+7)*Binomial(n+6, 6)/7 ); # G. C. Greubel, Jan 19 2020

Magma

[(10*n+7)*Binomial(n+6, 6)/7: n in [0..30]]; // G. C. Greubel, Jan 19 2020

Maple

seq( (10*n+7)*binomial(n+6, 6)/7, n=0..30); # G. C. Greubel, Jan 19 2020

Mathematica

Table[10*Binomial[n+7,7] -9*Binomial[n+6,6], {n,0,30}] (* G. C. Greubel, Jan 19 2020 *)
Rest[Nest[Accumulate[#]&,Table[n(n+1)(10n-7)/6,{n,0,50}],4]] (* Harvey P. Dale, Aug 03 2020 *)

PARI

vector(31, n, (10*n-3)*binomial(n+5, 6)/7) \\ G. C. Greubel, Jan 19 2020

Sage

[(10*n+7)*binomial(n+6, 6)/7 for n in (0..30)] # G. C. Greubel, Jan 19 2020

Formula

a(n) = (10*n + 7)*binomial(n+6, 6)/7.

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

From G. C. Greubel, Jan 19 2020: (Start)

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

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)

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