A322734 - OEIS

%I #11 Jan 10 2019 23:02:47

%S 1,1,13,301,13049,916441,94195333,13347584069,2494336502897,

%T 594306468307633,175843898741580413,63256176039805178717,

%U 27187896853252573010537,13760130793027073955444361,8099868218813257097451686389,5486947453030516571774107669621,4238147510641905386674640667028193,3702563543334459672543167863851104609,3632508681950057312629014573578376827629,3976725158932698015861683248077453527809421

%N Row sums of triangle A322731.

%H Paul D. Hanna, <a href="/A322734/b322734.txt">Table of n, a(n) for n = 0..49</a>

%F E.g.f. A(x) = C(x,y=x) given C(x,y) = 1 + Integral S(x,y)*C(y,x) dx such that C(x,y)^2 - S(x,y)^2 = 1 and C(y,x) = Integral S(y,x)*C(x,y) dy, where A(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)!.

%e E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 301*x^6/6! + 13049*x^8/8! + 916441*x^10/10! + 94195333*x^12/12! + 13347584069*x^14/14! + 2494336502897*x^16/16! + 594306468307633*x^18/18! + ...

%e RELATED SERIES.

%e sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 56*x^5/5! + 1856*x^7/7! + 103936*x^9/9! + 8893952*x^11/11! + 1080485888*x^13/13! + 176673603584*x^15/15! + 37417114009600*x^17/17! + 9963927777050624*x^19/19! + ... + A322733(n)*x^(2*n+1)/(2*n+1)! + ...

%e A(x) = cosh( Integral D(x) dx ) where D(x) = A'(x)/sqrt(A(x)^2 - 1) begins

%e D(x) = 1 + 3*x^2/2! + 25*x^4/4! + 595*x^6/6! + 26193*x^8/8! + 1832611*x^10/10! + 188365801*x^12/12! + 26696014003*x^14/14! + 4988672502305*x^16/16! + 1188611267890243*x^18/18! + ...

%o (PARI) {A322731(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)! *polcoeff(polcoeff(Cx, 2*n-2*k, x), 2*k, y)}

%o a(n) = sum(k=0, n, A322731(n, k))

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A322731, A322733.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 01 2019