A322733 - OEIS

%I #18 Mar 06 2019 13:26:44

%S 1,4,56,1856,103936,8893952,1080485888,176673603584,37417114009600,

%T 9963927777050624,3258468236083331072,1283800576046355447808,

%U 599766792781193235398656,327833344685493011492110336,207273007354010852132258840576,150092604050210703675500571656192,123411833893721992920958314267803648,114350314285874476606422143072852770816,118597403963878781240540181376561622024192,136853838009919098121511181384398866938331136

%N Row sums of triangle A322730.

%H Paul D. Hanna, <a href="/A322733/b322733.txt">Table of n, a(n) for n = 0..49</a>

%F E.g.f. A(x) = S(x,y=x) given 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, where A(x) = Sum_{n>=0} a(n) * x^(2*n+1)/(2*n+1)!.

%e E.g.f.: A(x) = 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! + ...

%e RELATED SERIES.

%e sqrt(1 + A(x)^2) = 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! + ... + A322734(n)*x^(2*n)/(2*n)! + ...

%e A(x) = sinh( Integral D(x) dx ) where D(x) = A'(x)/sqrt(1 + A(x)^2) 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) {A322730(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)! *polcoeff(polcoeff(Sx, 2*n+1-2*k, x), 2*k, y)}

%o a(n) = sum(k=0, n, A322730(n, k))

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

%Y Cf. A322730, A322734.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 01 2019