A053112 Expansion of (-1 + 1/(1-9*x)^9)/(81*x); related to A053108.

1, 45, 1485, 40095, 938223, 19702683, 379980315, 6839645670, 116273976390, 1883638417518, 29282015399598, 439230230993970, 6385731819835410, 90312492880529370, 1246312401751305306, 16825217423642621631, 222686701195269992175, 2894927115538509898275, 37024594161887258172675

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (81,-2916,61236,-826686,7440174,-44641044,172186884,-387420489,387420489).

Formula

G.f.: (-1 + 1/(1-9*x)^9)/(81*x).

a(n) = 9^(n-1)*binomial(n+9, 8).

a(n) = 81*a(n-1) - 2916*a(n-2) + 61236*a(n-3) - 826686*a(n-4) + 7440174*a(n-5) - 44641044*a(n-6) + 172186884*a(n-7) - 387420489*a(n-8) + 387420489*a(n-9), with a(0)=1, a(1)=45, a(2)=1485, a(3)=40095, a(4)=938223, a(5)=19702683, a(6)=379980315, a(7)=6839645670, a(8)=116273976390. - Harvey P. Dale, Apr 27 2013

From Amiram Eldar, Nov 03 2025: (Start)

Sum_{n>=0} 1/a(n) = 50419462581/35 - 12230590464*log(9/8).

Sum_{n>=0} (-1)^n/a(n) = 43012376919/7 - 58320000000*log(10/9). (End)

Mathematica

CoefficientList[Series[(-1+1/(1-9*x)^9)/(81*x), {x, 0, 30}], x] (* or *) LinearRecurrence[{81, -2916, 61236, -826686, 7440174, -44641044, 172186884, -387420489, 387420489}, {1, 45, 1485, 40095, 938223, 19702683, 379980315, 6839645670, 116273976390}, 20] (* Harvey P. Dale, Apr 27 2013 *)
Table[9^(n - 1)*Binomial[n + 9, 8], {n, 0, 30}] (* G. C. Greubel, Aug 16 2018 *)

Prog

(PARI) vector(30, n, n--; 9^(n-1)*binomial(n+9, 8)) \\ G. C. Greubel, Aug 16 2018
(Magma) [9^(n-1)*Binomial(n+9, 8): n in [0..30]]; // G. C. Greubel, Aug 16 2018

Crossrefs

Cf. A053108, A053110, A053111, A053113.

Without signs: A078812. With zeros: A049310.

Cf. A008310 (T(n, x)), A008312 (U(n, x)).

Sequence in context: A137716 A035521 A107399 * A240686 A014940 A273436

Adjacent sequences: A053109 A053110 A053111 * A053113 A053114 A053115

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved