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 A326796 E.g.f. C(x) = C(x,x), where C(x,y) is the e.g.f. of triangle A326799. 3

%I #4 Aug 06 2019 06:17:24

%S 2,-4,-154,1336,139402,-2516812,-531298962,16291452784,5792759106578,

%T -269619415772564,-144868583759257994,9514326895639572136,

%U 7196653593954009006746,-633707576374342516652764,-641861736344903371972973954,72970336686693711886017561312,95350631737851404765088770262050,-13593601759174079774108596113815332,-22265118012504823914985759717325911674

%N E.g.f. C(x) = C(x,x), where C(x,y) is the e.g.f. of triangle A326799.

%C Equals the row sums of triangle A326799.

%F E.g.f. C(x) and related functions A(x) and B(x), defined by A326794 and A326795, 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.: C(x) = 2*x^2/2! - 4*x^4/4! - 154*x^6/6! + 1336*x^8/8! + 139402*x^10/10! - 2516812*x^12/12! - 531298962*x^14/14! + 16291452784*x^16/16! + 5792759106578*x^18/18! - 269619415772564*x^20/20! + ...

%o (PARI) {a(n, k) = my(Ax=x, Bx=1, Cx=x, Ay=y, 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+2)! * polcoeff( polcoeff(Cx, 2*n-2*k+1, x), 2*k+1, y))}

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

%Y Cf. A326794, A326795, A326799.

%K sign

%O 2,1

%A _Paul D. Hanna_, Aug 05 2019

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