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A292124
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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.
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5
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6, 49, 336, 3608, 39984, 568456, 8788224, 157119872, 3070910976, 66631838464, 1568964182016, 40143982850048, 1104238186174464, 32575797728674816, 1024499039235538944, 34244972509718085632, 1211785683371470749696, 45266987105529337544704, 1779867330217651806928896, 73484865365862823153369088, 3178359543535472461866860544, 143720321925324472350632574976
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OFFSET
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0,1
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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