A056001 a(n) = (n+1)*binomial(n+7, 7).

1, 16, 108, 480, 1650, 4752, 12012, 27456, 57915, 114400, 213928, 381888, 655044, 1085280, 1744200, 2728704, 4167669, 6229872, 9133300, 13156000, 18648630, 26048880, 35897940, 48859200, 65739375, 87512256, 115345296, 150629248, 195011080, 250430400, 319159632

Offset: 0

Author

Barry E. Williams, Jun 18 2000

Keywords

• nonn
• easy

Comments

Original name: A second-order recursive sequence.

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Partial sums of A052226.

Cf. A093565 ((8, 1) Pascal, column m=8).

Cf. A007318, A000142, A245334.

Programs

GAP

List([0..30], n-> (n+1)*Binomial(n+7,7)); # G. C. Greubel, Aug 29 2019

Haskell

a056001 n = (n + 1) * a007318' (n + 7) 7
-- Reinhard Zumkeller, Aug 31 2014

Magma

[(n+1)*Binomial(n+7,7): n in [0..30]]; // G. C. Greubel, Aug 29 2019

Maple

seq((n+1)*binomial(n+7,7), n=0..30); # G. C. Greubel, Aug 29 2019

Mathematica

Table[(n+1)Binomial[n+7, 7], {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011; corrected by Bruno Berselli, Jan 23 2015 *)

PARI

vector(30, n, n*binomial(n+6,7)) \\ G. C. Greubel, Aug 29 2019

Sage

[(n+1)*binomial(n+7,7) for n in (0..30)] # G. C. Greubel, Aug 29 2019

Formula

G.f.: (1+7*x)/(1-x)^9.

a(n) = A245334(n+7,7)/A000142(7). - Reinhard Zumkeller, Aug 31 2014

a(n) = A000581(n+8)+7*A000581(n+7). - R. J. Mathar, Oct 24 2014

E.g.f.: (5040 +75600*x +194040*x^2 +170520*x^3 +66150*x^4 +12642*x^5 + 1225*x^6 +57*x^7 +x^8)*exp(x)/5040. - G. C. Greubel, Aug 29 2019

From Amiram Eldar, Jan 15 2023: (Start)

Sum_{n>=0} 1/a(n) = 7*Pi^2/6 - 37583/3600.

Sum_{n>=0} (-1)^n/a(n) = 7*Pi^2/12 - 2912*log(2)/15 + 155701/1200. (End)