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A140520
a(n) = binomial(n+9, 9)*5^n.
1
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
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
KEYWORD
nonn,easy
AUTHOR
Zerinvary Lajos, Jul 02 2008
STATUS
approved