A140520 a(n) = binomial(n+9, 9)*5^n.

1, 50, 1375, 27500, 446875, 6256250, 78203125, 893750000, 9496093750, 94960937500, 902128906250, 8201171875000, 71760253906250, 607202148437500, 4987731933593750, 39901855468750000, 311733245849609375, 2383842468261718750, 17878818511962890625, 131738662719726562500

Offset

0,2

Comments

With a different offset, number of n-permutations (n>=9) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly nine (9) u's.

Example: a(1)=50 because we have

uuuuuuuuut, uuuuuuuuuv, uuuuuuuuuz, uuuuuuuuux, uuuuuuuuuy,

uuuuuuuutu, uuuuuuuuvu, uuuuuuuuzu, uuuuuuuuxu, uuuuuuuuyu,

uuuuuuutuu, uuuuuuuvuu, uuuuuuuzuu, uuuuuuuxuu, uuuuuuuyuu,

uuuuuutuuu, uuuuuuvuuu, uuuuuuzuuu, uuuuuuxuuu, uuuuuuyuuu,

uuuuutuuuu, uuuuuvuuuu, uuuuuzuuuu, uuuuuxuuuu, uuuuuyuuuu,

uuuutuuuuu, uuuuvuuuuu, uuuuzuuuuu, uuuuxuuuuu, uuuuyuuuuu,

uuutuuuuuu, uuuvuuuuuu, uuuzuuuuuu, uuuxuuuuuu, uuuyuuuuuu,

uutuuuuuuu, uuvuuuuuuu, uuzuuuuuuu, uuxuuuuuuu, uuyuuuuuuu,

utuuuuuuuu, uvuuuuuuuu, uzuuuuuuuu, uxuuuuuuuu. uyuuuuuuuu,

tuuuuuuuuu, vuuuuuuuuu, zuuuuuuuuu, xuuuuuuuuu. yuuuuuuuuu.

Links

Table of n, a(n) for n=0..19.

Index entries for linear recurrences with constant coefficients, signature (50,-1125,15000,-131250,787500,-3281250,9375000,-17578125,19531250,-9765625).

Formula

From Chai Wah Wu, Mar 20 2017: (Start)

a(n) = 50*a(n-1) - 1125*a(n-2) + 15000*a(n-3) - 131250*a(n-4) + 787500*a(n-5) - 3281250*a(n-6) + 9375000*a(n-7) - 17578125*a(n-8) + 19531250*a(n-9) - 9765625*a(n-10) for n > 9.

G.f.: 1/(1 - 5*x)^10. (End)

From Amiram Eldar, Aug 29 2022: (Start)

Sum_{n>=0} 1/a(n) = 2949120*log(5/4) - 36852261/56.

Sum_{n>=0} (-1)^n/a(n) = 75582720*log(6/5) - 771700059/56. (End)

Maple

seq(binomial(n+9, 9)*5^n, n=0..20);

Mathematica

Table[Binomial[n + 9, 9] 5^n, {n, 0, 16}] (* or *)
CoefficientList[Series[1/(1 - 5 x)^10, {x, 0, 16}], x] (* Michael De Vlieger, Mar 20 2017 *)

Crossrefs

Sequence in context: A283311 A031602 A231743 * A114251 A004351 A229316

Adjacent sequences: A140517 A140518 A140519 * A140521 A140522 A140523

Keyword

nonn,easy

Author

Zerinvary Lajos, Jul 02 2008

Status

approved