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A286798
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Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. .
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4
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1, 1, 4, 2, 27, 22, 248, 264, 30, 2830, 3610, 830, 8, 38232, 55768, 18746, 1078, 593859, 961740, 414720, 46986, 576, 10401712, 18326976, 9457788, 1593664, 62682, 112, 202601898, 382706674, 226526362, 49941310, 3569882, 45296, 4342263000, 8697475368, 5740088706, 1540965514, 160998750, 4909674, 16896, 101551822350, 213865372020, 154271354280, 48205014786, 6580808784, 337737294, 4200032, 2560
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OFFSET
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0,3
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COMMENTS
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Row n>0 contains floor(2*(n+1)/3) terms.
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LINKS
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FORMULA
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y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies x^2*deriv(y,x) = (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2), with y(0;t) = 1, where P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n>0.
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EXAMPLE
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A(x;t) = 1 + x + (4 + 2*t)*x^2 + (27 + 22*t)*x^3 + (248 + 264*t + 30*t^2)*x^4 +
Triangle starts:
n\k [0] [1] [2] [3] [4] [5]
[0] 1;
[1] 1;
[2] 4, 2;
[3] 27, 22;
[4] 248, 264, 30;
[5] 2830, 3610, 830, 8;
[6] 38232, 55768, 18746, 1078;
[7] 593859, 961740, 414720, 46986, 576;
[8] 10401712, 18326976, 9457788, 1593664, 62682, 112;
[9] 202601898, 382706674, 226526362, 49941310, 3569882, 45296;
[10] ...
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MATHEMATICA
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max = 12; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = 1 + x y0[x, t]^2 + 3 t x^3 y0[x, t]^2 D[y0[x, t], x] + x^2 (2 y0[x, t] D[y0[x, t], x] + t (2 y0[x, t]^3 - D[y0[x, t], x] + y0[x, t] D[y0[x, t], x])) + O[x]^n // Normal // Simplify; y0[x_, t_] = y1[x, t]];
P[n_, t_] := Coefficient[y0[x, t] , x, n];
row[n_] := CoefficientList[P[n, t], t];
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PROG
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(PARI)
my(x='x+O('x^N), y0=1, y1=0, n=1);
while(n++,
y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1; ); y0;
};
my(v = A286795_ser(N, t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
};
concat(apply(p->Vecrev(p), Vec(A286798_ser(12))))
\\ test: y=A286798_ser(50); x^2*y' == (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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