A084950: Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.

Matches ?

Row sums ? A001040

Submission text ?

Triangle begins:
  0:       1;
  1:       2,       1;
  2:       6,       4;
  3:      24,      18,      1;
  4:     120,      96,      9;
  5:     720,     600,     72,     1;
  6:    5040,    4320,    600,    16;
  7:   40320,   35280,   5400,   200,   1;
  8:  362880,  322560,  52920,  2400,  25;
  9: 3628800, 3265920, 564480, 29400, 450, 1;

The row polynomials satisfy P(n,x) = (n+1)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(n + 1)*P[n - 1, x] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1)*v[i - 1] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
  by Liu–Wang theorem for three-term recurrences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Liu–Wang theorem for three-term recurrences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Liu–Wang theorem for three-term recurrences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
  by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

• Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 10

• shift_x_plus_one of Array of coefficients of denominator polynomials of the n-t… Image under a basis map u^k -> P_k (A007318)
• basis_map_A013610 of Array of coefficients of denominator polynomials of the n-… Image under a basis map u^k -> P_k (A013610)
• basis_map_A013609 of Array of coefficients of denominator polynomials of the n-… Image under a basis map u^k -> P_k (A013609)
• basis_map_A013611 of Array of coefficients of denominator polynomials of the n-… Image under a basis map u^k -> P_k (A013611)
• basis_map_A008277 of Array of coefficients of denominator polynomials of the n-… Image under a basis map u^k -> P_k (A008277)
• shift_x_plus_one of Array of coefficients of denominator polynomials of the n-t… Image under a real-rootedness preserver (affine, shift 0)
• derivative of Array of coefficients of denominator polynomials of the n-th appr… Image under a real-rootedness preserver (derivative polynomial, shift 0)
• theta_plus_one of Array of coefficients of denominator polynomials of the n-th … Image under a real-rootedness preserver (euler polynomial, shift 0)

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003