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

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A053108, A053110 - A053113.

Without signs: A078812. With zeros: A049310. Cf. A008310 (T(n, x)), A008312 (U(n, x)).

Programs

Magma

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

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 *)

PARI

vector(30,n,n--; 9^(n-1)*binomial(n+9, 8)) \\ G. C. Greubel, Aug 16 2018

Formula

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

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

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, 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). - Harvey P. Dale, Apr 27 2013