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A101850 A Catalan transform of Pell(n+1).

%I A101850 #44 Mar 15 2025 23:28:54
%S A101850 1,2,7,26,100,392,1555,6218,25006,100988,409162,1661948,6764194,
%T A101850 27575732,112570675,460058906,1881978694,7704907724,31566153058,
%U A101850 129400608044,530734613920,2177792579072,8939838222718,36711025334948
%N A101850 A Catalan transform of Pell(n+1).
%C A101850 A Catalan transform of the Pell numbers A000129(n+1) under the mapping G(x) -> G(x*c(x)), c(x) the g.f. of A000108. The inverse mapping is H(x) -> H(x*(1-x)).
%C A101850 Hankel transform is 3^n. - _Paul Barry_, Jan 19 2009
%C A101850 Row sums of the Riordan matrix (1/(x+sqrt(1-4*x)),(1-sqrt(1-4*x))/(2*(x+sqrt(1-4*x)))) (A188513). - _Emanuele Munarini_, Apr 02 2011
%C A101850 Equals the INVERT transform of A026641: (1, 1, 4, 13, 46, 166, ...). Example: a(4) = 100 = (1, 1, 2, 7, 26) dot (46, 13, 4, 1, 1) = (46 + 13 + 8 + 7 + 26 ) = 100. - _Gary W. Adamson_, Jan 10 2012
%H A101850 G. C. Greubel, <a href="/A101850/b101850.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from Vincenzo Librandi)
%H A101850 Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012
%F A101850 G.f.: 2/(3*sqrt(1-4*x) + 2*x - 1).
%F A101850 a(n) = Sum_{k=0..n} (k/(2*n-k))*binomial(2*n-k, n-k)*A000129(k+1).
%F A101850 a(n) = Sum_{k=0..n} A039599(n,k)*A016116(k). - _Philippe Deléham_, Oct 29 2008
%F A101850 G.f.: 1/(1-2*x-3*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-... (continued fraction). - _Paul Barry_, Jan 19 2009
%F A101850 From _Emanuele Munarini_, Apr 02 2011: (Start)
%F A101850 a(n) = [x^n] (1-2*x)/((1-2*x-x^2)(1-x)^(n+1)).
%F A101850 a(n) = Sum_{k=0..n} binomial(2*n+1, n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2).
%F A101850 Recurrence: 2*(n+3)*a(n+3) -4*(4*n+9)*a(n+2) +(31*n+45)*a(n+1) +2*(2*n+3)*a(n) = 0. (End)
%F A101850 From _Gary W. Adamson_, Jul 14 2011: (Start)
%F A101850 a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:
%F A101850   2, 3, 0, 0, 0, 0, ...
%F A101850   1, 1, 1, 0, 0, 0, ...
%F A101850   1, 1, 1, 1, 0, 0, ...
%F A101850   1, 1, 1, 1, 1, 0, ...
%F A101850   1, 1, 1, 1, 1, 1, ...
%F A101850   ... (End)
%F A101850 a(n) ~ (1/4)*(2 + 3/sqrt(2))^n. - _Vaclav Kotesovec_, Oct 17 2012
%t A101850 CoefficientList[Series[(1-2x+3Sqrt[1-4x])/(4-16x-2x^2),{x,0,24}],x] (* _Emanuele Munarini_, Apr 02 2011 *)
%t A101850 CoefficientList[Series[2/(3*Sqrt[1-4*x]+2*x-1), {x,0,30}], x] (* or *) RecurrenceTable[{2*(n+3)*a[n+3] -4*(4*n+9)*a[n+2]+(31*n+45)*a[n+1] + 2*(2*n+3)*a[n]==0, a[0]==1, a[1]==2, a[2]==7}, a, {n, 0, 30}](* _G. C. Greubel_, Feb 18 2019 *)
%o A101850 (Maxima) makelist(sum(binomial(2*n+1,n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2),k,0,n),n,0,12); /* _Emanuele Munarini_, Apr 02 2011 */
%o A101850 (PARI) my(x='x+O('x^30)); Vec(2/(3*sqrt(1-4*x)+2*x-1)) \\ _G. C. Greubel_, Feb 18 2019
%o A101850 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2/(3*Sqrt(1-4*x)+2*x-1) )); // _G. C. Greubel_, Feb 18 2019
%o A101850 (Sage) (2/(3*sqrt(1-4*x)+2*x-1)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 18 2019
%Y A101850 Cf. A081696, A026641, A188513.
%K A101850 easy,nonn
%O A101850 0,2
%A A101850 _Paul Barry_, Dec 18 2004