A326794 - OEIS

%I #5 Aug 06 2019 06:17:13

%S 1,-4,-24,832,14080,-1178112,-40198144,6102368256,350164353024,

%T -84145181818880,-7290125908443136,2544780720155394048,

%U 310189596697035800576,-148294858002251055628288,-24194060487469916561080320,15177404660097357616248782848,3193403421862605489912894455808,-2544493850062379218318289711136768,-670888183682611536585868983069048832,661838176807194100001000065515567185920

%N E.g.f. A(x) = A(x,x), where A(x,y) is the e.g.f. of triangle A326797.

%C Equals the row sums of triangle A326797.

%F E.g.f. A(x) and related functions B(x) and C(x), defined by A326795 and A326796, respectively, satisfy:

%F (1) A(x)^2 + B(x)^2 + C(x)^2 = 1,

%F (2) A(x)*A'(x) + B(x)*B'(x) + C(x)*C'(x) = 0.

%e E.g.f.: A(x) = x - 4*x^3/3! - 24*x^5/5! + 832*x^7/7! + 14080*x^9/9! - 1178112*x^11/11! - 40198144*x^13/13! + 6102368256*x^15/15! + 350164353024*x^17/17! - 84145181818880*x^19/19! + ...

%o (PARI) {a(n) = my(Ax=1, Bx=x, Cx=1, Ay=1, By=y, Cy=1);

%o for(i=0, 2*n+1,

%o Ax = 0 + intformal( Bx*Cy - Cx*By, x) + O(x^(2*n+2));

%o Bx = 1 + intformal( Cx*Ay - Ax*Cy, x) + O(x^(2*n+2));

%o Cx = 0 + intformal( Ax*By - Bx*Ay, x) + O(x^(2*n+2));

%o Ay = 0 + intformal( By*Cx - Cy*Bx, y) + O(y^(2*n+2));

%o By = 0 + intformal( Cy*Ax - Ay*Cx, y) + O(y^(2*n+2));

%o Cy = 1 + intformal( Ay*Bx - By*Ax, y) + O(y^(2*n+2));

%o );

%o sum(k=0,n, (2*n+1)! * polcoeff( polcoeff(Ax, 2*n-2*k+1, x), 2*k, y))}

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

%Y Cf. A326795, A326796, A326797.

%K sign

%O 1,2

%A _Paul D. Hanna_, Aug 05 2019