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

0,2

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).

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 +22861440*x^2 +26853120*x^3 +14605920*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)

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 *)

Prog

(Haskell)
a056114 n = (n + 1) * a007318' (n + 9) 9
-- Reinhard Zumkeller, Aug 31 2014
(PARI) vector(41, n, n*binomial(n+8, 9) ) \\ G. C. Greubel, Jan 18 2020
(Magma) [(n+1)*Binomial(n+9, 9): n in [0..40]]; // G. C. Greubel, Jan 18 2020
(Sage) [(n+1)*binomial(n+9, 9) for n in (0..40)] # G. C. Greubel, Jan 18 2020
(GAP) List([0..40], n-> (n+1)*Binomial(n+9, 9)); # G. C. Greubel, Jan 18 2020

Crossrefs

Cf. A000142, A007318, A056003, A245334.

Sequence in context: A041768 A221870 A289181 * A281204 A186259 A292281

Adjacent sequences: A056111 A056112 A056113 * A056115 A056116 A056117

Keyword

nonn,easy

Author

Barry E. Williams, Jun 12 2000

Status

approved