A053113 - OEIS

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, 3124550000000000000000, 46868250000000000000000

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OFFSET

0,2

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

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.

Index entries for linear recurrences with constant coefficients, signature (100,-4500,120000,-2100000,25200000,-210000000,1200000000,-4500000000,10000000000,-10000000000).

FORMULA

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

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

From Amiram Eldar, Nov 03 2025: (Start)

Sum_{n>=0} 1/a(n) = 387420489000*log(10/9) - 285731757475/7.

Sum_{n>=0} (-1)^n/a(n) = 1287126759925/7 - 1929229929000*log(11/10). (End)

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 *)

PROG

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

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

CROSSREFS

Cf. A000012, A001792, A036068, A036070, A036083, A036224, A053110, A053111, A053112.

Sequence in context: A383840 A381584 A240687 * A012048 A215860 A020536

Adjacent sequences: A053110 A053111 A053112 * A053114 A053115 A053116

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved