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A141407
a(n) = binomial(n+7,7)*6^n.
4
1, 48, 1296, 25920, 427680, 6158592, 80061696, 960740352, 10808328960, 115288842240, 1175946190848, 11545653510144, 109683708346368, 1012465000120320, 9112185001082880, 80187228009529344, 691614841582190592, 5858384540460908544, 48819871170507571200, 400836836978904268800
OFFSET
0,2
COMMENTS
With a different offset, number of n-permutations (n>=7) of 7 objects: s, t, u, v, z, x, y with repetition allowed, containing exactly seven (7) u's.
If n=7 then a(0)= 1 because we have uuuuuuu
a(1)=48 because we have
uuuuuuus, uuuuuuut, uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuuy,
uuuuuusu, uuuuuutu, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuuyu,
uuuuusuu, uuuuutuu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuuyuu,
uuuusuuu, uuuutuuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuuyuuu,
uuusuuuu, uuutuuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuuyuuuu,
uusuuuuu, uutuuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uuyuuuuu,
usuuuuuu, utuuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, uyuuuuuu,
suuuuuuu, tuuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu, yuuuuuuu.
LINKS
Index entries for linear recurrences with constant coefficients, signature (48,-1008,12096,-90720,435456,-1306368,2239488,-1679616).
FORMULA
O.g.f.: 1/(1-6x)^8. - R. J. Mathar, Aug 08 2008
a(n) = 48*a(n-1) - 1008*a(n-2) + 12096*a(n-3) - 90720*a(n-4) + 435456*a(n-5) - 1306368*a(n-6) + 2239488*a(n-7) - 1679616*a(n-8) for n > 7. - Chai Wah Wu, Nov 12 2021
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 656250*log(6/5) - 239295/2.
Sum_{n>=0} (-1)^n/a(n) = 4941258*log(7/6) - 7616973/10. (End)
MAPLE
seq(binomial(n+7, 7)*6^n, n=0..16);
MATHEMATICA
Table[Binomial[n + 7, 7] 6^n, {n, 0, 16}] (* Michael De Vlieger, Jul 24 2017 *)
PROG
(Magma) [6^n* Binomial(n+7, 7): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011
(PARI) vector(15, n, binomial(n+6, 7)*6^(n-1)) \\ Derek Orr, Jul 24 2017
CROSSREFS
Cf. A038255.
Sequence in context: A285169 A163272 A165283 * A004341 A222199 A288459
KEYWORD
nonn,easy
AUTHOR
Zerinvary Lajos, Aug 04 2008
STATUS
approved