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

0,2

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)

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

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 *)

Prog

(PARI) vector(31, n, (10*n-3)*binomial(n+5, 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
(Sage) [(10*n+7)*binomial(n+6, 6)/7 for n in (0..30)] # G. C. Greubel, Jan 19 2020
(GAP) List([0..30], n-> (10*n+7)*Binomial(n+6, 6)/7 ); # G. C. Greubel, Jan 19 2020

Crossrefs

Cf. A050406.

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

Sequence in context: A080441 A135400 A372583 * A156851 A354183 A141921

Adjacent sequences: A052251 A052252 A052253 * A052255 A052256 A052257

Keyword

easy,nonn

Author

Barry E. Williams, Feb 03 2000

Status

approved