A053113 Expansion of (-1 + 1/(1-10*x)^10)/(100*x); related to A053109.

1, 55, 2200, 71500, 2002000, 50050000, 1144000000, 24310000000, 486200000000, 9237800000000, 167960000000000, 2939300000000000, 49742000000000000, 817190000000000000, 13075040000000000000, 204297500000000000000

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-1)*binomial(n+k,k-1) starting with A000012 (powers of 1), A001792, A036068, A036070, A036083, A036224, A053110-113 for k=1..10.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
• Index entries for linear recurrences with constant coefficients, signature (100, -4500, 120000, -2100000, 25200000, -210000000, 1200000000, -4500000000, 10000000000, -10000000000).

Programs

Magma

[10^(n-1)*Binomial(n+10, 9): n in [0..30]]; // G. C. Greubel, Aug 16 2018

Mathematica

Table[10^(n - 1)*Binomial[n + 10, 9], {n, 0, 30}] (* G. C. Greubel, Aug 16 2018 *)
LinearRecurrence[{100,-4500,120000,-2100000,25200000,-210000000,1200000000,-4500000000,10000000000,-10000000000},{1,55,2200,71500,2002000,50050000,1144000000,24310000000,486200000000,9237800000000},20] (* Harvey P. Dale, Jul 30 2025 *)

PARI

vector(30,n,n--; 10^(n-1)*binomial(n+10, 9)) \\ G. C. Greubel, Aug 16 2018

Formula

a(n) = 10^(n-1)*binomial(n+10, 9).

G.f.: (-1 + (1-10*x)^(-10))/(x*10^2).