A053464 a(n) = n*5^(n-1).

0, 1, 10, 75, 500, 3125, 18750, 109375, 625000, 3515625, 19531250, 107421875, 585937500, 3173828125, 17089843750, 91552734375, 488281250000, 2593994140625, 13732910156250, 72479248046875, 381469726562500

Offset

0,3

Comments

With a different offset, number of n-permutations of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly one u. - Zerinvary Lajos, Dec 28 2007

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Frank Ellermann, Illustration of binomial transforms

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 756.

Index entries for linear recurrences with constant coefficients, signature (10,-25).

Formula

a(n) = Sum_{k=0..n} 5^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2. - Paul Barry, Oct 15 2004

a(n) = 10*a(n-1) - 25*a(n-2); n>1; a(0)=0, a(1)=1.

Fourth binomial transform of n (starting 0, 1, 10...) Convolution of powers of 5.

G.f.: x/(1-5*x)^2; E.g.f.: x*exp(5*x). - Paul Barry, Jul 22 2003

a(n) = - 25^n * a(-n) for all n in Z. - Michael Somos, Jun 26 2017

From Amiram Eldar, Oct 28 2020: (Start)

Sum_{n>=1} 1/a(n) = 5*log(5/4).

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(6/5). (End)

Mathematica

Join[{a=0, b=1}, Table[c=10*b-25*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
Table[n*5^(n-1), {n, 0, 20}] (* or *) LinearRecurrence[{10, -25}, {0, 1}, 30] (* Harvey P. Dale, Jul 22 2014 *)

Prog

(PARI) {a(n) = n*5^(n-1)}; /* Michael Somos, Sep 12 2005 */
(Sage) [lucas_number1(n, 10, 25) for n in range(0, 21)] # Zerinvary Lajos, Apr 26 2009
(Magma) [n*(5^(n-1)): n in [0..30]]; // Vincenzo Librandi, Jun 09 2011

Crossrefs

Cf. A002697, A027471, A001787.

Sequence in context: A027203 A291003 A305784 * A111998 A026935 A110127

Adjacent sequences: A053461 A053462 A053463 * A053465 A053466 A053467

Keyword

easy,nonn

Author

Barry E. Williams, Jan 13 2000

Extensions

More terms from James A. Sellers, Feb 02 2000

Status

approved