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%I #25 Sep 24 2017 21:29:49
%S 1,0,0,0,-1,1,0,0,0,0,-2,4,-2,0,0,0,1,-3,2,-8,15,-7,0,0,0,0,10,-40,30,
%T 10,20,-50,20,0,0,0,-1,11,0,34,-304,594,-634,520,-281,61,0,0,0,0,-42,
%U 364,-462,168,-2296,4956,-2436,-952,378,504,-182,0,0,0,1,-43,-138,668,5857,-27669,49144,-50908,42411,-36853,29530,-16368,4915,-547,0,0,0,0,170,-3280,6110,-6830,115840,-369410,430280,-255040,233510,-223880,19450,84310,-28120,-4750,1640,0,0,0,-1,171,1900,-22482,-121338,1032888,-2402754,2941956,-4315854,8677464,-11619696,8799414,-4061298,1757028,-1018098,428240,-82461,4921,0,0,0
%N E.g.f. C = C(x,y) satisfies: A^2 + B^2 + C^2 = 1 + y^2 and A^3 + B^3 + C^3 = 1 + y^3, where functions A = A(x,y) and B = B(x,y) are described by A278885 and A278886, respectively.
%C This triangle is a row reversal of triangle A278886.
%H Paul D. Hanna, <a href="/A278887/b278887.txt">Table of n, a(n) for n = 0..1680 of rows 0..40 of the triangle in flattened form.</a>
%F Given e.g.f. C(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=1..2*n+1} T(n,k)*y^k, functions A = A(x,y), B = B(x,y), and C = C(x,y) satisfy:
%F (1) A^2 + B^2 + C^2 = 1 + y^2,
%F (2) A^3 + B^3 + C^3 = 1 + y^3,
%F where y is a parameter independent of x.
%F Vector [A,B,C] equals the integration of the cross product specified by:
%F (3) [A,B,C] = [0,1,y] + Integral [A,B,C] X [A^2,B^2,C^2] dx,
%F thus [A',B',C'] is orthogonal to [A,B,C] and [A^2,B^2,C^2].
%F Explicitly,
%F (3.a) A = Integral B*C^2 - B^2*C dx,
%F (3.b) B = 1 + Integral C*A^2 - C^2*A dx,
%F (3.c) C = y + Integral A*B^2 - A^2*B dx.
%F Since [A',B',C'] = [A,B,C] X [A^2,B^2,C^2], then
%F (4) A'^2 + B'^2 + C'^2 = (1+y^2)*(A^4 + B^4 + C^4) - (1+y^3)^2.
%F (5) [A',B',C'] X [A,B,C] = (1+y^2)*[A^2,B^2,C^2] - (1+y^3)*[A,B,C];
%F explicitly,
%F (5.a) B'*C - C'*B = (1+y^2)*A^2 - (1+y^3)*A,
%F (5.b) C'*A - A'*C = (1+y^2)*B^2 - (1+y^3)*B,
%F (5.c) A'*B - B'*A = (1+y^2)*C^2 - (1+y^3)*C.
%F Let D = A^4 + B^4 + C^4, then
%F (6) [A^2,B^2,C^2] X [A',B',C'] = D*[A,B,C] - (1+y^3)*[A^2,B^2,C^2];
%F explicitly,
%F (6.a) B^2*C' - C^2*B' = D*A - (1+y^3)*A^2,
%F (6.b) C^2*A' - A^2*C' = D*B - (1+y^3)*B^2,
%F (6.c) A^2*B' - B^2*A' = D*C - (1+y^3)*C^2.
%e This triangle of coefficients T(n,k) of x^n*y^k/n! in C(x,y), for n>=0, k=1..2*n+1, begins:
%e 1;
%e 0, 0, 0;
%e -1, 1, 0, 0, 0;
%e 0, -2, 4, -2, 0, 0, 0;
%e 1, -3, 2, -8, 15, -7, 0, 0, 0;
%e 0, 10, -40, 30, 10, 20, -50, 20, 0, 0, 0;
%e -1, 11, 0, 34, -304, 594, -634, 520, -281, 61, 0, 0, 0;
%e 0, -42, 364, -462, 168, -2296, 4956, -2436, -952, 378, 504, -182, 0, 0, 0;
%e 1, -43, -138, 668, 5857, -27669, 49144, -50908, 42411, -36853, 29530, -16368, 4915, -547, 0, 0, 0;
%e 0, 170, -3280, 6110, -6830, 115840, -369410, 430280, -255040, 233510, -223880, 19450, 84310, -28120, -4750, 1640, 0, 0, 0;
%e -1, 171, 1900, -22482, -121338, 1032888, -2402754, 2941956, -4315854, 8677464, -11619696, 8799414, -4061298, 1757028, -1018098, 428240, -82461, 4921, 0, 0, 0; ...
%e where C(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=1..2*n+1} T(n,k)*y^k.
%e ...
%e E.g.f.: C(x,y) = y + (y^2 - y)*x^2/2! + (-2*y^4 + 4*y^3 - 2*y^2)*x^3/3! +
%e (-7*y^6 + 15*y^5 - 8*y^4 + 2*y^3 - 3*y^2 + y)*x^4/4! +
%e (20*y^8 - 50*y^7 + 20*y^6 + 10*y^5 + 30*y^4 - 40*y^3 + 10*y^2)*x^5/5! +
%e (61*y^10 - 281*y^9 + 520*y^8 - 634*y^7 + 594*y^6 - 304*y^5 + 34*y^4 + 11*y^2 - y)*x^6/6! +
%e (-182*y^12 + 504*y^11 + 378*y^10 - 952*y^9 - 2436*y^8 + 4956*y^7 - 2296*y^6 + 168*y^5 - 462*y^4 + 364*y^3 - 42*y^2)*x^7/7! +...
%e such that functions A = A(x,y), B = B(x,y), and C = C(x,y) satisfy:
%e (1) A^2 + B^2 + C^2 = 1 + y^2 and
%e (2) A^3 + B^3 + C^3 = 1 + y^3.
%e RELATED SERIES.
%e A(x,y) = (y^2 - y)*x + (-y^6 + 3*y^5 - 2*y^4 + 2*y^3 - 3*y^2 + y)*x^3/3! + (2*y^7 - 8*y^6 + 6*y^5 + 6*y^4 - 8*y^3 + 2*y^2)*x^4/4! + (y^10 - 11*y^9 + 20*y^8 - 44*y^7 + 104*y^6 - 104*y^5 + 44*y^4 - 20*y^3 + 11*y^2 - y)*x^5/5! + (-10*y^11 + 100*y^10 - 150*y^9 - 70*y^8 + 130*y^7 + 130*y^6 - 70*y^5 - 150*y^4 + 100*y^3 - 10*y^2)*x^6/6! + (-y^14 + 43*y^13 - 142*y^12 + 466*y^11 - 2245*y^10 + 5423*y^9 - 7480*y^8 + 7480*y^7 - 5423*y^6 + 2245*y^5 - 466*y^4 + 142*y^3 - 43*y^2 + y)*x^7/7! +...
%e B(x,y) = 1 + (-y^4 + y^3)*x^2/2! + (2*y^5 - 4*y^4 + 2*y^3)*x^3/3! + (y^8 - 3*y^7 + 2*y^6 - 8*y^5 + 15*y^4 - 7*y^3)*x^4/4! + (-10*y^9 + 40*y^8 - 30*y^7 - 10*y^6 - 20*y^5 + 50*y^4 - 20*y^3)*x^5/5! + (-y^12 + 11*y^11 + 34*y^9 - 304*y^8 + 594*y^7 - 634*y^6 + 520*y^5 - 281*y^4 + 61*y^3)*x^6/6! + (42*y^13 - 364*y^12 + 462*y^11 - 168*y^10 + 2296*y^9 - 4956*y^8 + 2436*y^7 + 952*y^6 - 378*y^5 - 504*y^4 + 182*y^3)*x^7/7! +...
%e The scalar triple product yields
%e [A',B',C'] * ([A,B,C] X [A^2,B^2,C^2]) = A'^2 + B'^2 + C'^2
%e where
%e A'^2 + B'^2 + C'^2 = (y^4 - 2*y^3 + y^2) + (4*y^7 - 8*y^6 + 8*y^5 - 8*y^4 + 4*y^3)*x^2/2! + (-8*y^9 + 16*y^8 - 8*y^7 + 8*y^5 - 16*y^4 + 8*y^3)*x^3/3! +
%e (-28*y^11 + 124*y^10 - 240*y^9 + 356*y^8 - 424*y^7 + 356*y^6 - 240*y^5 + 124*y^4 - 28*y^3)*x^4/4! + (80*y^13 - 200*y^12 - 120*y^11 + 480*y^10 - 200*y^9 + 200*y^7 - 480*y^6 + 120*y^5 + 200*y^4 - 80*y^3)*x^5/5! + (244*y^15 - 2148*y^14 + 7048*y^13 - 13684*y^12 + 20236*y^11 - 25128*y^10 + 26864*y^9 - 25128*y^8 + 20236*y^7 - 13684*y^6 + 7048*y^5 - 2148*y^4 + 244*y^3)*x^6/6! +...
%e Also, we have the relation
%e A'^2 + B'^2 + C'^2 = (1+y^2)*(A^4 + B^4 + C^4) - (1+y^3)^2
%e where
%e A^4 + B^4 + C^4 = (y^4 + 1) + (4*y^5 - 8*y^4 + 4*y^3)*x^2/2! + (-8*y^7 + 16*y^6 - 16*y^4 + 8*y^3)*x^3/3! + (-28*y^9 + 124*y^8 - 212*y^7 + 232*y^6 - 212*y^5 + 124*y^4 - 28*y^3)*x^4/4! + (80*y^11 - 200*y^10 - 200*y^9 + 680*y^8 - 680*y^6 + 200*y^5 + 200*y^4 - 80*y^3)*x^5/5! + (244*y^13 - 2148*y^12 + 6804*y^11 - 11536*y^10 + 13432*y^9 - 13592*y^8 + 13432*y^7 - 11536*y^6 + 6804*y^5 - 2148*y^4 + 244*y^3)*x^6/6! +...
%o (PARI) {T(n,k) = my(A=x, B=1, C=y); for(i=1, n,
%o A = intformal(B*C^2 - B^2*C +x*O(x^n));
%o B = 1 + intformal(C*A^2 - C^2*A);
%o C = y + intformal(A*B^2 - A^2*B); ); polcoeff( n!*polcoeff(C, n, x), k, y)}
%o for(n=0,10, for(k=1,2*n+1, print1(T(n,k),", "));print(""))
%Y Cf. A278885 (A(x,y)), A278886 (B(x,y)), A278889 (central terms).
%Y Cf. A278746 (A at y=2), A278747 (B at y=2), A278748 (C at y=2).
%K sign,tabf
%O 0,11
%A _Paul D. Hanna_, Dec 19 2016
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