A025250 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 0, 1, 1, 0.
0, 1, 1, 0, 1, 1, 1, 3, 3, 6, 11, 15, 31, 50, 85, 161, 267, 490, 883, 1548, 2863, 5127, 9307, 17116, 31021, 57123, 104963, 192699, 356643, 658034, 1218517, 2262079, 4196895, 7812028, 14549655, 27126118, 50671255, 94697293, 177220411, 332015747
Offset: 1
Keywords
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
- Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
- Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
- Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Programs
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GAP
List([0..45], n-> Sum([0..n], k-> Binomial(k+1,n-2*k-1)*Binomial(n-k-2,k)/(k+1) )); # G. C. Greubel, Feb 23 2019
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Haskell
a025250 n = a025250_list !! (n-1) a025250_list = 0 : 1 : 1 : f 1 [1,1,0] where f k xs = x' : f (k+1) (x':xs) where x' = sum $ zipWith (*) a025250_list $ take k xs -- Reinhard Zumkeller, Nov 03 2011 -
Magma
m:=45; R
:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( (1 +x^2 -Sqrt(1-2*x^2-4*x^3+x^4))/2 )); // G. C. Greubel, Feb 23 2019 -
Mathematica
Rest[CoefficientList[Series[(1+x^2-Sqrt[1-2*x^2-4*x^3+x^4])/2, {x,0,40}],x]] (* Harvey P. Dale, Apr 05 2011 *) Rest@CoefficientList[Series[x^2+ContinuedFractionK[-x^3,x^2-1,{k, 0, 40}],{x,0,40}], x] (* Benedict W. J. Irwin, Oct 13 2016 *) -
Maxima
a(n):=sum((binomial(k+1,n-2*k-1)*binomial(n-k-2,k))/(k+1),k,0,n); /* Vladimir Kruchinin, Nov 22 2014 */
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PARI
a(n)=polcoeff((x^2-sqrt(1-2*x^2-4*x^3+x^4+x*O(x^n)))/2,n)
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Sage
a=((1+x^2 -sqrt(1-2*x^2-4*x^3+x^4))/2).series(x, 45).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 23 2019
Formula
G.f.: (1 +x^2 -sqrt(1-2*x^2-4*x^3+x^4))/2. - Michael Somos, Jun 08 2000
G.f.: x^2+x^3*(1/(1-x^2))c(x^3/(1-x^2)^2) where c(x) is the g.f. of A000108. - Paul Barry, May 20 2009
a(n+2) = Sum_{k=0..n} binomial((n+k)/2, 2*k)*(1+(-1)^(n-k))*A000108(k)/2. - Paul Barry, Jul 06 2009
a(n) = Sum_{k=0..n} binomial(k+1,n-2*k-1)*binomial(n-k-2,k)/(k+1). - Vladimir Kruchinin, Nov 22 2014
G.f.: K_{k>=0} (-x^3)/(x^2-1), where K is the Gauss notation for a continued fraction. - Benedict W. J. Irwin, Oct 11 2016
a(n) ~ sqrt(1 - r^2 - r^3) * (2*r + 4*r^2 - r^3)^n / (2*sqrt(Pi)*n^(3/2)), where r = 0.51361982956383128341133963576515885989214886200017191578885... is the root of the equation 1 - 2*r^2 - 4*r^3 + r^4 = 0. - Vaclav Kotesovec, Jul 03 2021
Shifts left 3 places under the INVERT transform. - J. Conrad, Mar 08 2023
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