A056114 Expansion of (1+9*x)/(1-x)^11.

1, 20, 165, 880, 3575, 12012, 35035, 91520, 218790, 486200, 1016158, 2015520, 3821090, 6963880, 12257850, 20920064, 34730575, 56241900, 89049675, 138138000, 210315105, 314757300, 463681725, 673171200, 964177500, 1363732656, 1906401420, 2636011840, 3607704980

Offset: 0

Author

Barry E. Williams, Jun 12 2000

Keywords

• nonn
• easy

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Crossrefs

Cf. A000142, A007318, A056003, A245334.

Programs

GAP

List([0..40], n-> (n+1)*Binomial(n+9, 9)); # G. C. Greubel, Jan 18 2020

Haskell

a056114 n = (n + 1) * a007318' (n + 9) 9
-- Reinhard Zumkeller, Aug 31 2014

Magma

[(n+1)*Binomial(n+9, 9): n in [0..40]]; // G. C. Greubel, Jan 18 2020

Maple

a:=n->(sum((numbcomp(n,10)), j=10..n)):seq(a(n), n=10..34); # Zerinvary Lajos, Aug 26 2008

Mathematica

CoefficientList[Series[(1+9x)/(1-x)^11,{x,0,40}],x] (* or *) LinearRecurrence[ {11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,20,165,880,3575,12012,35035, 91520,218790,486200,1016158},40] (* Harvey P. Dale, Jun 05 2018 *)

PARI

vector(41, n, n*binomial(n+8, 9) ) \\ G. C. Greubel, Jan 18 2020

Sage

[(n+1)*binomial(n+9, 9) for n in (0..40)] # G. C. Greubel, Jan 18 2020

Formula

a(n) = (n+1)*binomial(n+9, 9).

G.f.: (1+9*x)/(1-x)^11.

a(n) = A245334(n+9,9)/A000142(9). - Reinhard Zumkeller, Aug 31 2014

From G. C. Greubel, Jan 18 2020: (Start)

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

E.g.f.: (9! +6894720*x [phone]*x^2 [phone]*x^3 [phone]*x^4 + 4191264*x^5 +677376*x^6 +63072*x^7 +3321*x^8 +91*x^9 +x^10)*exp(x)/9!. (End)

From Amiram Eldar, Jan 15 2023: (Start)

Sum_{n>=0} 1/a(n) = 3*Pi^2/2 - 1077749/78400.

Sum_{n>=0} (-1)^n/a(n) = 3*Pi^2/4 - 24576*log(2)/35 + 37652469/78400. (End)