A278885 E.g.f. A = A(x,y) satisfies: A^2 + B^2 + C^2 = 1 + y^2 and A^3 + B^3 + C^3 = 1 + y^3, where functions B = B(x,y) and C = C(x,y) are described by A278886 and A278887, respectively.

-1, 1, 0, 0, 0, 0, 1, -3, 2, -2, 3, -1, 0, 2, -8, 6, 6, -8, 2, 0, -1, 11, -20, 44, -104, 104, -44, 20, -11, 1, 0, -10, 100, -150, -70, 130, 130, -70, -150, 100, -10, 0, 1, -43, 142, -466, 2245, -5423, 7480, -7480, 5423, -2245, 466, -142, 43, -1, 0, 42, -1008, 2646, -462, 4704, -23730, 17808, 17808, -23730, 4704, -462, 2646, -1008, 42, 0, -1, 171, -1040, 3888, -45138, 215718, -501504, 720816, -790524, 790524, -720816, 501504, -215718, 45138, -3888, 1040, -171, 1, 0, -170, 9500, -42150, 38990, -422070, 2104870, -3396830, 1821030, -113170, -113170, 1821030, -3396830, 2104870, -422070, 38990, -42150, 9500, -170, 0

Offset

1,8

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1640 of rows 1..40 of the triangle in flattened form.

Formula

Given e.g.f. A(x,y) = Sum_{n>=1} x^n/n! * Sum_{k=1..2*n} T(n,k)*y^k, functions A = A(x,y), B = B(x,y), and C = C(x,y) satisfy:

(1) A^2 + B^2 + C^2 = 1 + y^2,

(2) A^3 + B^3 + C^3 = 1 + y^3,

where y is a parameter independent of x.

Vector [A,B,C] equals the integration of the cross product specified by:

(3) [A,B,C] = [0,1,y] + Integral [A,B,C] X [A^2,B^2,C^2] dx,

thus [A',B',C'] is orthogonal to [A,B,C] and [A^2,B^2,C^2].

Explicitly,

(3.a) A = Integral B*C^2 - B^2*C dx,

(3.b) B = 1 + Integral C*A^2 - C^2*A dx,

(3.c) C = y + Integral A*B^2 - A^2*B dx.

Since [A',B',C'] = [A,B,C] X [A^2,B^2,C^2], then

(4) A'^2 + B'^2 + C'^2 = (1+y^2)*(A^4 + B^4 + C^4) - (1+y^3)^2.

(5) [A',B',C'] X [A,B,C] = (1+y^2)*[A^2,B^2,C^2] - (1+y^3)*[A,B,C];

explicitly,

(5.a) B'*C - C'*B = (1+y^2)*A^2 - (1+y^3)*A,

(5.b) C'*A - A'*C = (1+y^2)*B^2 - (1+y^3)*B,

(5.c) A'*B - B'*A = (1+y^2)*C^2 - (1+y^3)*C.

Let D = A^4 + B^4 + C^4, then

(6) [A^2,B^2,C^2] X [A',B',C'] = D*[A,B,C] - (1+y^3)*[A^2,B^2,C^2];

explicitly,

(6.a) B^2*C' - C^2*B' = D*A - (1+y^3)*A^2,

(6.b) C^2*A' - A^2*C' = D*B - (1+y^3)*B^2,

(6.c) A^2*B' - B^2*A' = D*C - (1+y^3)*C^2.

ROW SUMS:

Sum_{k=1..2*n} T(n,k) = 0, for n>=1.

Sum_{k=1..4*n} k * T(2*n,k) = 0, for n>=1.

Sum_{k=1..4*n-2} k * T(2*n-1,k) = 2^(n-1), for n>=1.

Example

This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y), for n>=1, k=1..2*n, begins:
-1, 1;
0, 0, 0, 0;
1, -3, 2, -2, 3, -1;
0, 2, -8, 6, 6, -8, 2, 0;
-1, 11, -20, 44, -104, 104, -44, 20, -11, 1;
0, -10, 100, -150, -70, 130, 130, -70, -150, 100, -10, 0;
1, -43, 142, -466, 2245, -5423, 7480, -7480, 5423, -2245, 466, -142, 43, -1;
0, 42, -1008, 2646, -462, 4704, -23730, 17808, 17808, -23730, 4704, -462, 2646, -1008, 42, 0;
-1, 171, -1040, 3888, -45138, 215718, -501504, 720816, -790524, 790524, -720816, 501504, -215718, 45138, -3888, 1040, -171, 1;
0, -170, 9500, -42150, 38990, -422070, 2104870, -3396830, 1821030, -113170, -113170, 1821030, -3396830, 2104870, -422070, 38990, -42150, 9500, -170, 0; ...
where A(x,y) = Sum_{n>=1} x^n/n! * Sum_{k=1..2*n} T(n,k)*y^k.
...
E.g.f.: 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! +...
such that functions A = A(x,y), B = B(x,y), and C = C(x,y) satisfy:
(1) A^2 + B^2 + C^2 = 1 + y^2 and
(2) A^3 + B^3 + C^3 = 1 + y^3.
RELATED SERIES.
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! +...
C(x,y) = y + (y^2 - y)*x^2/2! + (-2*y^4 + 4*y^3 - 2*y^2)*x^3/3! + (-7*y^6 + 15*y^5 - 8*y^4 + 2*y^3 - 3*y^2 + y)*x^4/4! + (20*y^8 - 50*y^7 + 20*y^6 + 10*y^5 + 30*y^4 - 40*y^3 + 10*y^2)*x^5/5! + (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! + (-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! +...
The scalar triple product yields
[A',B',C'] * ([A,B,C] X [A^2,B^2,C^2]) = A'^2 + B'^2 + C'^2
where
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! +
(-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! +...
Also, we have the relation
A'^2 + B'^2 + C'^2 = (1+y^2)*(A^4 + B^4 + C^4) - (1+y^3)^2
where
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! +...

Prog

(PARI) {T(n, k) = my(A=x, B=1, C=y); for(i=1, n,
A = intformal(B*C^2 - B^2*C +x*O(x^n));
B = 1 + intformal(C*A^2 - C^2*A);
C = y + intformal(A*B^2 - A^2*B); ); polcoeff( n!*polcoeff(A, n, x), k, y)}
for(n=1, 10, for(k=1, 2*n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A278886 (B(x,y)), A278887 (C(x,y)), A278888 (central terms).

Cf. A278746 (A at y=2), A278747 (B at y=2), A278748 (C at y=2).

Sequence in context: A105198 A133882 A092106 * A183049 A178086 A353296

Adjacent sequences: A278882 A278883 A278884 * A278886 A278887 A278888

Keyword

sign,tabf

Author

Paul D. Hanna, Dec 19 2016

Status

approved