A292124 E.g.f. D(x) = A(x)*B(x)*C(x) where A(x), B(x), and C(x) are the e.g.f.s of A292121, A292122, and A292123, respectively.

6, 49, 336, 3608, 39984, 568456, 8788224, 157119872, 3070910976, 66631838464, 1568964182016, 40143982850048, 1104238186174464, 32575797728674816, 1024499039235538944, 34244972509718085632, 1211785683371470749696, 45266987105529337544704, 1779867330217651806928896, 73484865365862823153369088, 3178359543535472461866860544, 143720321925324472350632574976

Offset

0,1

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

E.g.f. D(x) = A(x)*B(x)*C(x) and related functions A(x), B(x), and C(x) satisfy:

(1a) D(x) = A(x)*A'(x).

(1b) D(x) = B(x)*B'(x).

(1c) D(x) = C(x)*C'(x).

(2a) A(x)^2 = 1 + Integral 2*D(x) dx.

(2b) B(x)^2 = 4 + Integral 2*D(x) dx.

(2c) C(x)^2 = 9 + Integral 2*D(x) dx.

(3a) A(x) = 1 + Integral B(x)*C(x) dx.

(3b) B(x) = 2 + Integral A(x)*C(x) dx.

(3c) C(x) = 3 + Integral A(x)*B(x) dx.

(4a) B(x)^2 - A(x)^2 = 3.

(4b) C(x)^2 - A(x)^2 = 8.

(4c) C(x)^2 - B(x)^2 = 5.

(5a) A(x)^m = 1 + Integral m * D(x) * A(x)^(m-2) dx.

(5b) B(x)^m = 2^m + Integral m * D(x) * B(x)^(m-2) dx.

(5c) C(x)^m = 3^m + Integral m * D(x) * C(x)^(m-2) dx.

Example

E.g.f. D(x) = 6 + 49*x + 336*x^2/2! + 3608*x^3/3! + 39984*x^4/4! + 568456*x^5/5! + 8788224*x^6/6! + 157119872*x^7/7! + 3070910976*x^8/8! + 66631838464*x^9/9! + 1568964182016*x^10/10! +...
where D(x) = A(x)*B(x)*C(x).
Related series.
A(x) = 1 + 6*x + 13*x^2/2! + 102*x^3/3! + 653*x^4/4! + 7134*x^5/5! + 80257*x^6/6! + 1138638*x^7/7! + 17577977*x^8/8! + 314204406*x^9/9! + 6141247573*x^10/10! +...
where A(x)*A'(x) = D(x).
B(x) = 2 + 3*x + 20*x^2/2! + 78*x^3/3! + 736*x^4/4! + 6672*x^5/5! + 83360*x^6/6! + 1113072*x^7/7! + 17810944*x^8/8! + 311847168*x^9/9! + 6167567360*x^10/10! +...
where B(x)*B'(x) = D(x).
C(x) = 3 + 2*x + 15*x^2/2! + 82*x^3/3! + 759*x^4/4! + 6698*x^5/5! + 83355*x^6/6! + 1111018*x^7/7! + 17804811*x^8/8! + 311922962*x^9/9! + 6167999175*x^10/10! +...
where C(x)*C'(x) = D(x).

Prog

(PARI) {a(n) = my(A=1, B=2, C=3); for(i=0, n, A = 1 + intformal(B*C +x*O(x^n)); B = 2 + intformal(A*C); C = 3 + intformal(A*B)); n!*polcoeff(A*B*C, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A292120 (A+B+C), A292121 (A), A292122 (B), A292123 (C).

Sequence in context: A027766 A097299 A283226 * A104170 A098306 A055847

Adjacent sequences: A292121 A292122 A292123 * A292125 A292126 A292127

Keyword

nonn

Author

Paul D. Hanna, Sep 08 2017

Status

approved