A103365 - OEIS

%I #18 Apr 01 2018 07:53:55

%S 1,-1,2,-7,39,-321,3681,-56197,1102571,-27036487,810263398,

%T -29139230033,1238451463261,-61408179368043,3513348386222286,

%U -229724924077987509,17023649385410772579,-1419220037471837658603,132236541042728184852942,-13690229149108218523467549

%N First column of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

%H G. C. Greubel, <a href="/A103365/b103365.txt">Table of n, a(n) for n = 1..250</a>

%F From _Paul D. Hanna_, Jan 31 2009: (Start)

%F G.f.: A(x) = 1/B(x) where A(x) = Sum_{n>=0} (-1)^n*a(n)*x^n/[n!*(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n].

%F G.f. satisfies: A(x) = 1/F(x*A(x)) and F(x) = 1/A(x*F(x)) where F(x) = Sum_{n>=0} A155926(n)*x^n/[n!*(n+1)!/2^n].

%F G.f. satisfies: A(x) = 1/G(x/A(x)) and G(x) = 1/A(x/G(x)) where G(x) = Sum_{n>=0} A155927(n)*x^n/[n!*(n+1)!/2^n]. (End)

%F a(n) ~ (-1)^(n+1) * c * n! * (n-1)! * d^n, where d = 4/BesselJZero[1, 1]^2 = 0.2724429913055159309179376055957891881897555639652..., and c = 9.11336321311226744479181866135367355200240221549667284076... = BesselJZero[1, 1]^2 / (4*BesselJ[2, BesselJZero[1, 1]]). - _Vaclav Kotesovec_, Mar 01 2014, updated Apr 01 2018

%e From _Paul D. Hanna_, Jan 31 2009: (Start)

%e G.f.: A(x) = 1 - x + 2*x^2/3 - 7*x^3/18 + 39*x^4/180 - 321*x^5/2700 +...

%e G.f.: A(x) = 1/B(x) where:

%e B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +... (End)

%t Table[(-1)^((n-1)/2) * (CoefficientList[Series[x/BesselJ[1,2*x],{x,0,40}],x])[[n]] * ((n+1)/2)! * ((n-1)/2)!,{n,1,41,2}] (* _Vaclav Kotesovec_, Mar 01 2014 *)

%o (PARI) a(n)=if(n<1,0,(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^-1)[n,1])

%o (PARI) {a(n)=local(B=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(1/B,n)*n!*(n+1)!/2^n} \\ _Paul D. Hanna_, Jan 31 2009

%Y Cf. A001263, A103364, A103366, A103367.

%Y Cf. A155926, A155927. [_Paul D. Hanna_, Jan 31 2009]

%Y Cf. A238390, A180874, A115369.

%K sign

%O 1,3

%A _Paul D. Hanna_, Feb 02 2005