A325153 - OEIS

%I #18 Apr 04 2019 12:19:20

%S 1,5,33,277,2465,22149,199297,1793621,16142529,145282693,1307544161,

%T 11767897365,105911076193,953199685637,8578797170625,77209174535509,

%U 694882570819457,6253943137374981,56285488236374689,506569394127372053,4559124547146348321,41032120924317134725,369289088318854212353,3323601794869687910997,29912416153827191198785,269211745384444720788869

%N A column of triangle A322220; a(n) = A322220(n,1) for n >= 1.

%C The e.g.f. of triangle A322220 is S(x,y) = Integral C(x,y)*C(y,x) dx such that C(x,y)^2 - S(x,y)^2 = 1 and C(y,x) = 1 + Integral S(y,x)*C(x,y) dy.

%F a(n) = (9^n - 9)/24 + n, for n >= 1.

%F E.g.f.: (8 - 3*(3 - 8*x)*exp(x) + exp(9*x))/24.

%F O.g.f.: x*(1 - 6*x - 3*x^2) / ((1 - x)^2 * (1 - 9*x)).

%e E.g.f.: A(x) = x + 5*x^2/2! + 33*x^3/3! + 277*x^4/4! + 2465*x^5/5! + 22149*x^6/6! + 199297*x^7/7! + 1793621*x^8/8! + 16142529*x^9/9! + 145282693*x^10/10! + ...

%e which equals (8 - (9 - 24*x)*exp(x) + exp(9*x))/24.

%e RELATED TRIANGLE AND SERIES.

%e Triangle A322220 of coefficients T(n,k) of x^(2*n+1-2*k)*y^(2*k)/((2*n+1-2*k)!*(2*k)!) in S(x,y) starts as follows:

%e 1;

%e 1, 1;

%e 1, 5, 1;

%e 1, 33, 33, 1;

%e 1, 277, 561, 277, 1;

%e 1, 2465, 10545, 10545, 2465, 1;

%e 1, 22149, 220065, 368213, 220065, 22149, 1;

%e 1, 199297, 4983681, 13530881, 13530881, 4983681, 199297, 1;

%e 1, 1793621, 118758993, 532981813, 799527361, 532981813, 118758993, 1793621, 1; ...

%e in which this sequence forms a column and diagonal.

%e The related series S(x,y) begins as

%e S(x,y) = x + (1*x^3/3! + 1*x*y^2/2!) + (1*x^5/5! + 5*x^3*y^2/(3!*2!) + 1*x*y^4/4!) + (1*x^7/7! + 33*x^5*y^2/(5!*2!) + 33*x^3*y^4/(3!*4!) + 1*x*y^6/6!) + (1*x^9/9! + 277*x^7*y^2/(7!*2!) + 561*x^5*y^4/(5!*4!) + 277*x^3*y^6/(3!*6!) + 1*x*y^8/8!) + (1*x^11/11! + 2465*x^9*y^2/(9!*2!) + 10545*x^7*y^4/(7!*4!) + 10545*x^5*y^6/(5!*6!) + 2465*x^3*y^8/(3!*8!) + 1*x*y^10/10!) + (1*x^13/13! + 22149*x^11*y^2/(11!*2!) + 220065*x^9*y^4/(9!*4!) + 368213*x^7*y^6/(7!*6!) + 220065*x^5*y^8/(5!*8!) + 22149*x^3*y^10/(3!*10!) +1*x*y^12/12!) + ...

%e and is defined by

%e S(x,y) = Integral C(x,y)*C(y,x) dx  such that C(x,y)^2 = 1 + S(x,y)^2.

%e Also, related series C(x,y) begins with

%e C(x,y) = 1 + 1*x^2/2! + (1*x^4/4! + 2*x^2*y^2/(2!*2!)) + (1*x^6/6! + 12*x^4*y^2/(4!*2!) + 8*x^2*y^4/(2!*4!)) + (1*x^8/8! + 94*x^6*y^2/(6!*2!) + 136*x^4*y^4/(4!*4!) + 32*x^2*y^6/(2!*6!)) + (1*x^10/10! + 824*x^8*y^2/(8!*2!) + 2400*x^6*y^4/(6!*4!) + 1760*x^4*y^6/(4!*6!) + 128*x^2*y^8/(2!*8!)) + (1*x^12/12! + 7386*x^10*y^2/(10!*2!) + 47600*x^8*y^4/(8!*4!) + 62096*x^6*y^6/(6!*6!) + 25728*x^4*y^8/(4!*8!) + 512*x^2*y^10/(2!*10!)) + ...

%e and may be defined by

%e C(x,y) = cosh( Integral C(y,x) dx ), and

%e C(y,x) = cosh( Integral C(x,y) dy ).

%o (PARI) {a(n) = (9^n - 9)/24 + n}

%o for(n=1, 30, print1( a(n), ", "));

%o (PARI) {A322220(n, k) = my(Sx=x, Sy=y, Cx=1, Cy=1); for(i=1, 2*n,

%o Sx = intformal( Cx*Cy +x*O(x^(2*n)), x);

%o Cx = 1 + intformal( Sx*Cy, x);

%o Sy = intformal( Cy*Cx +y*O(y^(2*k)), y);

%o Cy = 1 + intformal( Sy*Cx, y));

%o (2*n+1-2*k)!*(2*k)! *polcoeff(polcoeff(Sx, 2*n+1-2*k, x), 2*k, y)}

%o {a(n) = A322220(n, 1)}

%o for(n=1, 30, print1( a(n), ", "));

%Y Cf. A322220.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Apr 03 2019