A121965 a(n) = (n-1)*a(n-1)-a(n-2), a(0)=0, a(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
Offset: 0
Keywords
Examples
G.f. = x + x^2 + x^3 + 2*x^4 + 7*x^5 + 33*x^6 + 191*x^7 + 1304*x^8 + ...
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.
Programs
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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 *) RecurrenceTable[{a[0]==0,a[1]==1,a[n]==(n-1)a[n-1]-a[n-2]},a,{n,30}] (* Harvey P. Dale, Jul 13 2019 *) -
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 */
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
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
Comments