A053109 Expansion of 1/(1-10*x)^10.

1, 100, 5500, 220000, 7150000, 200200000, 5005000000, 114400000000, 2431000000000, 48620000000000, 923780000000000, 16796000000000000, 293930000000000000, 4974200000000000000, 81719000000000000000

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

This is the tenth member of the k-family of sequences a(k,n) := k^n*binomial(n+k-1,k-1) starting with A000012 (powers of 1), A001787(n+1), A027472(n+3), A038846, A036071, A036084, A036226, A053107-9 for k=1..10.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (100, -4500, 120000, -2100000, 25200000, -210000000, 1200000000, -4500000000, 10000000000, -10000000000).

Programs

GAP

List([0..15],n->10^n*Binomial(n+9,9)); # Muniru A Asiru, Aug 16 2018

Magma

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

Maple

seq(coeff(series(1/(1-10*x)^10, x, n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Aug 16 2018

Mathematica

CoefficientList[Series[1/(1-10x)^10,{x,0,20}],x] (* or *) Table[10^n Binomial[n+9,9],{n,0,20}] (* Harvey P. Dale, May 19 2011 *)

PARI

vector(30,n,n--; 10^n*binomial(n+9, 9)) \\ G. C. Greubel, Aug 16 2018

Sage

[lucas_number2(n, 10, 0)*binomial(n,9)/10 ^9 for n in range(9, 24)] # Zerinvary Lajos, Mar 13 2009

Formula

a(n) = 10^n*binomial(n+9, 9);

G.f.: 1/(1-10*x)^10.