A092766 Triangle read by rows: coefficients of Yablonskii-Vorob'ev polynomials.

1, 1, 1, -1, 1, -5, -5, 1, -15, 0, -175, 1, -35, 175, -1225, -12250, 6125, 1, -70, 1155, -9800, -67375, -1414875, 4716250, 2358125, 1, -126, 4725, -80850, 242550, -12733875, -202327125, 3034906875, 0, 11802415625, 1, -210, 15015, -512050, 7882875, -121396275, -1618617000, -24886236375, 1933235679375, -6750981737500, 35442654121875, 177213270609375, -59071090203125

Offset

0,6

Comments

Row n contains 1 + floor(n*(n+1)/6) terms (A008748). - Gheorghe Coserea, Nov 10 2016

Links

Gheorghe Coserea, Rows n = 0..44, flattened

M. Kaneko and H. Ochiai, On coefficients of Yablonskii-Vorob'ev polynomials

Formula

T(n) = Sum {k = 0..A008748(n)-1} a(n,k) * x^(A000217(n) - 3*k), where T(n)*T(n-2) = x*T(n-1)^2 + T(n-1)*T(n-1)'' - T(n-1)'^2, with T(0) = 1, T(1) = x. - Gheorghe Coserea, Nov 10 2016

Example

T(0) = 1,
T(1) = x,
T(2) = x^3 - 1,
T(3) = x^6 - 5*x^3 - 5,
T(4) = x^10 - 15*x^7 - 175*x,
T(5) = x^15 - 35*x^12 + 175*x^9 - 1225*x^6 - 12250*x^3 + 6125,
...
From Gheorghe Coserea, Nov 10 2016: (Start)
Triangle starts:
n\k [0]     [1]     [2]     [3]     [4]     [5]
[0]  1;
[1]  1;
[2]  1,     -1;
[3]  1,     -5,     -5;
[4]  1,    -15,      0,   -175;
[5]  1,    -35,    175,  -1225, -12250,   6125;
...
(End)

Mathematica

T[0][_] = 1; T[1][x_] := x; T[n_][x_] := T[n][x] = (x T[n-1][x]^2 + T[n-1][x] T[n-1]''[x] - T[n-1]'[x]^2)/T[n-2][x] // Simplify;
row[n_] := Join[{1}, Partition[CoefficientList[T[n][x], x] // Reverse // Rest, 3][[All, 3]]];
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Oct 23 2018 *)

Prog

(PARI) T(n)=if(n<2, if(n<1, n>=0, x), (x*T(n-1)^2+T(n-1)*T(n-1)''-T(n-1)'^2)/T(n-2))
(PARI)
seq(N) = {
  my(x = 'x, t = vector(N));
  t[1] = x; t[2] = x^3 - 1;
  for (n = 3, N,
    t[n] = (x*t[n-1]^2 + t[n-1]*t[n-1]'' - t[n-1]'^2)/t[n-2]);
  concat(1, t);
};
pol2row(p) = {
  my(tn = poldegree(p));
  vector(1 + tn\3, k, polcoeff(p, tn-3*(k-1)));
};
concat(apply(pol2row, seq(8))) \\ Gheorghe Coserea, Nov 10 2016

Crossrefs

Cf. A000217, A008748, A177893.

Sequence in context: A075298 A370262 A060058 * A288389 A060074 A280868

Adjacent sequences: A092763 A092764 A092765 * A092767 A092768 A092769

Keyword

sign,tabf

Author

Ralf Stephan, Apr 23 2004

Extensions

Offset corrected by Gheorghe Coserea, Nov 10 2016

Status

approved