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A109516 a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,n-1]. 0
1, 1, 6, 45, 464, 6000, 93528, 1707111, 35721216, 843160671, 22165100000, 642268811184, 20339749638144, 698946255836933, 25903663544572800, 1029945249481640625, 43733528272753917952, 1975222567881226040760 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;1,1] is the Fibonacci number A000045(n).

LINKS

Table of n, a(n) for n=1..18.

FORMULA

a(n) = [x^n] 1/(1 - n*x - n*x^2). - Paul D. Hanna, Dec 27 2012

a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*n^(n-k) for n>=0 (conjectured). - Werner Schulte, Oct 21 2016

a(n) = ((n + sqrt((n-1)*(n+3)) - 1)^n - (n - sqrt((n-1)*(n+3)) - 1)^n) / (2^n * sqrt((n-1)*(n+3))), for n > 1. - Daniel Suteu, Apr 20 2018

a(n) ~ n^(n-1). - Vaclav Kotesovec, Apr 20 2018

EXAMPLE

a(4)=45 because if M is the 2 X 2 matrix [0,1;3,3], then M^4 is the 2 X 2 matrix [36,45;135;171].

G.f. = x + x^2 + 6*x^3 + 45*x^4 + 464*x^5 + 6000*x^6 + 93528*x^7 + 1707111*x^8 + ...

MAPLE

with(linalg): a:=proc(n) local A, k: A[1]:=matrix(2, 2, [0, 1, n-1, n-1]): for k from 2 to n do A[k]:=multiply(A[k-1], A[1]) od: A[n][1, 2] end: seq(a(n), n=1..20);

MATHEMATICA

M[n_] = If[n > 1, MatrixPower[{{0, 1}, {n - 1, n - 1}}, n], {{0, 1}, {1, 1}}]; a = Table[M[n][[1, 2]], {n, 1, 50}]

Table[SeriesCoefficient[1/(1 - n*x - n*x^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 20 2018 *)

PROG

(PARI) {a(n)=polcoeff(1/(1-n*x-n*x^2+x*O(x^n)), n)} \\ Paul D. Hanna, Dec 27 2012

(PARI) a(n) = ([0, 1; n-1, n-1]^n)[1, 2]; \\ Michel Marcus, Apr 20 2018

CROSSREFS

Cf. A000045, A000166.

Sequence in context: A084064 A186925 A294642 * A245493 A078865 A160492

Adjacent sequences:  A109513 A109514 A109515 * A109517 A109518 A109519

KEYWORD

nonn

AUTHOR

Roger L. Bagula, Jun 16 2005

STATUS

approved

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Last modified September 25 19:05 EDT 2018. Contains 315394 sequences. (Running on oeis4.)