basis-map coefficients of A026729, shifted → basis A013609 → D+1

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Triangle begins:
   1:   1;
   2:  -3,   -2;
   3:   5,   12,     4;
   4:  -7,  -30,   -36,    -8;
   5:   9,   56,   120,    96,     16;
   6: -11,  -90,  -280,  -400,   -240,    -32;
   7:  13,  132,   540,  1120,   1200,    576,     64;
   8: -15, -182,  -924, -2520,  -3920,  -3360,  -1344,   -128;
   9:  17,  240,  1456,  4928,  10080,  12544,   8960,   3072,   256;
  10: -19, -306, -2160, -8736, -22176, -36288, -37632, -23040, -6912, -512;

The row polynomials satisfy (2*n-3+2*x)*P(n,x) = -(2*n-1+4*n*x+4*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + 4*n*x + 4*x^2)*P[n - 1, x])/(2*n - 3 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 1 + 4*n*x + 4*x^2)*v[i - 1])/(2*n - 3 + 2*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A156992 → basis A013611 → D+1
• A013609 → basis A013610 → D+1
• A134082 → basis A013609
• A136572 → basis A013609 → D+1
• A085881 → basis A013611 → D+1
• A178821 → basis A013611 → D+1
• A038208 → basis A013611 → D+1
• A120909 → basis A013610 → D+1
• A178822 → basis A013611 → D+1
• A013609 → D → D+1
• A085880 → basis A013611 → D+1
• A130123 → basis A013609 → D+1
• basis-map coefficients of A026729 → basis A013609 → D+1
• A038221 → basis A013611 → D+1
• basis-map coefficients of A030528, shifted → basis A013609 → D+1
• A196347 → basis A013611 → D+1
• basis-map coefficients of A112227 → basis A013609 → D+1
• A120909 → D+1
• A356546 → basis A013611 → D+1
• A178820 → basis A013611 → D+1
• A028326 → basis A013611 → D+1
• A013609 → D+1

References

1. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.