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A121965 a(n) = (n-1)*a(n-1)-a(n-2), a(0)=0, a(1)=1. 1
0, 1, 1, 1, 2, 7, 33, 191, 1304, 10241, 90865, 898409, 9791634, 116601199, 1506023953, 20967734143, 313009988192, 4987192076929, 84469255319601, 1515459403675889, 28709259414522290, 572669728886769911, 11997355047207645841, 263369141309681438591 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n+1) = A058797(n).

REFERENCES

Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, (1945), page 144.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..451

Isak Hilmarsson, Ingibjorg Jonsdottir, Steinunn Sigurdardottir and Sigridur Vidarsdottir, Wilf-classification of mesh patterns of short length, Reykjavík University, Thesis, May 2011.

FORMULA

a(n) = ( J_n(2)*Y_0(2) - J_0(2)*Y_n(2) )/( J_1(2)* Y_0(2) - J_0(2)*Y_1(2) ) where J and Y are Bessel functions.

a(-n) = (-1)^n * a(n). - Michael Somos, Jan 28 2014

0 = a(n)*(a(n+2)) + a(n+1)*(-a(n+1) + a(n+2) - a(n+3)) + a(n+2)*(a(n+2)) for all n in Z. - Michael Somos, Jan 28 2014

G.f. A(x) =: y satisfies y = x + x^2 * (y' - y) as formal power series. - Michael Somos, Jan 28 2014

E.g.f. A(x) =: y satisfies 0 = y''' * (y' - y) + y'' * (y' - 2*y''). - Michael Somos, Jan 28 2014

a(n+1) = Sum_{k=0..n/2} (-1)^k * (n-k)! / k! * binomial(n-k, k) if n>=-1. - Michael Somos, Jan 28 2014

EXAMPLE

G.f. = x + x^2 + x^3 + 2*x^4 + 7*x^5 + 33*x^6 + 191*x^7 + 1304*x^8 + ...

MATHEMATICA

Needs["DiscreteMath`RSolve`"]; Clear[f]; f[n_Integer] = Module[{a}, a[n] /.RSolve[{a[n] == (n - 1)*a[n - 1] - a[n - 2], a[0] == 0, a[1] == 1}, a[n], n][[1]] // Simplify] // ToRadicals Table[Floor[N[f[n]]], {n, 0, 25}]

a[ n_] := With[ {m = Abs[n]}, If[ n==0, 0, Sign[n]^m Sum[ (-1)^k (m - k)! / k! Binomial[m - k, k], {k, 0, m/2}]]]; (* Michael Somos, Jan 28 2014 *)

PROG

(PARI) {a(n) = local(s, A); s=sign(n); n=abs(n); if( n>0, A=vector(n, k, 1); for(k=4, n, A[k] = (k-1) * A[k-1] - A[k-2]); s^n * A[n], 0)}; /* Michael Somos, Jan 28 2014 */

(PARI) {a(n) = local(s, a0, a1, a2); s = sign(n); s^n * if( n!=0, a1=1; for( k=2, abs(n), a2 = (k-1) * a1 - a0; a0 = a1; a1 = a2); a1, 0)}; /* Michael Somos, Jan 28 2014 */

(PARI) {a(n) = local(s, A); s=sign(n); n=abs(n); A = O(x); for(k=1, n, A = (1 + x * A') * x / (1 + x^2)); s^n * polcoeff(A, n)}; /* Michael Somos, Jan 28 2014 */

(PARI) {a(n) = local(s); if( n==0, 0, s=sign(n); n=abs(n)-1; s^(n+1) * sum(k=0, n\2, (-1)^k * (n-k)! / k! * binomial(n-k, k)))}; /* Michael Somos, Jan 28 2014 */

CROSSREFS

Cf. A007754, A056921 (bisection), A058797, A106174.

Sequence in context: A162661 A104981 A058797 * A006595 A179525 A217033

Adjacent sequences:  A121962 A121963 A121964 * A121966 A121967 A121968

KEYWORD

nonn

AUTHOR

Roger L. Bagula, Sep 02 2006

EXTENSIONS

Values (with rounding errors) and offset corrected by the Assoc. Eds. of the OEIS, Mar 27 2010

Added a(0)=0 from Michael Somos, Jan 28 2014

STATUS

approved

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Last modified March 27 17:28 EDT 2017. Contains 284177 sequences.