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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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