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

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Triangle begins:
   1:  1;
   2: -1,  -4;
   3:  1,   8,   12;
   4: -1, -12,  -36,   -32;
   5:  1,  16,   72,   128,     80;
   6: -1, -20, -120,  -320,   -400,   -192;
   7:  1,  24,  180,   640,   1200,   1152,    448;
   8: -1, -28, -252, -1120,  -2800,  -4032,  -3136,  -1024;
   9:  1,  32,  336,  1792,   5600,  10752,  12544,   8192,   2304;
  10: -1, -36, -432, -2688, -10080, -24192, -37632, -36864, -20736, -5120;

The row polynomials satisfy (1+(2*n-2)*x)*P(n,x) = -(1+(2*n+2)*x+4*n*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[(-(1 + (2*n + 2)*x + 4*n*x^2)*P[n - 1, x])/(1 + (2*n - 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] = (-(1 + (2*n + 2)*x + 4*n*x^2)*v[i - 1])/(1 + (2*n - 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 ?

• A136572 → basis A013609 → θ+1
• A028326 → basis A013611 → θ+1
• A120909 → basis A013610 → θ+1
• A130123 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A013609 → θ+1
• A356546 → basis A013611 → θ+1
• A116666 → x ↦ x/(x+1)
• A196347 → basis A013611 → θ+1
• A085880 → basis A013611 → θ+1
• A038221 → basis A013611 → θ+1
• basis-map coefficients of A112227 → basis A013609 → θ+1
• A178820 → basis A013611 → θ+1
• A120909 → θ+1
• A038208 → basis A013611 → θ+1
• A013609 → basis A013610 → θ+1
• A013609 → D → θ+1
• A178821 → basis A013611 → θ+1
• basis-map coefficients of A030528, shifted → basis A013609 → θ+1
• A013609 → θ+1
• A085881 → basis A013611 → θ+1
• A134082 → basis A007318 → x ↦ x+1
• A156992 → basis A013611 → θ+1
• A116666 → reverse → x ↦ x+1
• A178822 → basis A013611 → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.