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A140220
a(n) = binomial(n+7, 7)*5^n.
2
1, 40, 900, 15000, 206250, 2475000, 26812500, 268125000, 2513671875, 22343750000, 189921875000, 1553906250000, 12301757812500, 94628906250000, 709716796875000, 5204589843750000, 37407989501953125, 264056396484375000, 1833724975585937500, 12546539306640625000
OFFSET
0,2
COMMENTS
With a different offset, number of n-permutations (n>=7) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly seven (7) u's.
If n=7 then a(0)=1.
Example: a(1)=40 because we have
uuuuuuut, uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuuy,
uuuuuutu, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuuyu,
uuuuutuu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuuyuu,
uuuutuuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuuyuuu,
uuutuuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuuyuuuu,
uutuuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uuyuuuuu,
utuuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, uyuuuuuu,
tuuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu, yuuuuuuu.
LINKS
Index entries for linear recurrences with constant coefficients, signature (40,-700,7000,-43750,175000,-437500,625000,-390625).
FORMULA
From Chai Wah Wu, Mar 20 2017: (Start)
a(n) = 40*a(n-1) - 700*a(n-2) + 7000*a(n-3) - 43750*a(n-4) + 175000*a(n-5) - 437500*a(n-6) + 625000*a(n-7) - 390625*a(n-8) for n > 7.
G.f.: 1/(1 - 5*x)^8. (End)
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 143360*log(5/4) - 191933/6.
Sum_{n>=0} (-1)^n/a(n) = 1632960*log(6/5) - 1786337/6. (End)
MAPLE
seq(binomial(n+7, 7)*5^n, n=0..18);
MATHEMATICA
Table[Binomial[n+7, 7]*5^n, {n, 0, 20}] (* Harvey P. Dale, Oct 07 2014 *)
CoefficientList[Series[1 / (1 - 5 x)^8, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 09 2018 *)
PROG
(Magma) [Binomial(n+7, 7)*5^n: n in [0..20]]; // Vincenzo Librandi, Feb 09 2018
CROSSREFS
Sequence in context: A013346 A013348 A013349 * A130610 A279582 A004339
KEYWORD
nonn,easy
AUTHOR
Zerinvary Lajos, Jun 23 2008
STATUS
approved