basis-map coefficients of A026729, shifted → basis A013609 → (θ+1)²

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Submission text ?

Triangle begins:
   1:  1;
   2: -1,  -8;
   3:  1,  16,    36;
   4: -1, -24,  -108,   -128;
   5:  1,  32,   216,    512,    400;
   6: -1, -40,  -360,  -1280,  -2000,   -1152;
   7:  1,  48,   540,   2560,   6000,    6912,    3136;
   8: -1, -56,  -756,  -4480, -14000,  -24192,  -21952,   -8192;
   9:  1,  64,  1008,   7168,  28000,   64512,   87808,   65536,   20736;
  10: -1, -72, -1296, -10752, -50400, -145152, -263424, -294912, -186624, -51200;

The row polynomials satisfy (1+(6*n-8)*x+(4*n^2-8*n+4)*x^2)*P(n,x) = -(1+6*n*x+(4*n^2+12*n-4)*x^2+8*n^2*x^3)*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 + 6*n*x + (4*n^2 + 12*n - 4)*x^2 + 8*n^2*x^3)*P[n - 1, x])/(1 + (6*n - 8)*x + (4*n^2 - 8*n + 4)*x^2)]];
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 + 6*n*x + (4*n^2 + 12*n - 4)*x^2 + 8*n^2*x^3)*v[i - 1])/(1 + (6*n - 8)*x + (4*n^2 - 8*n + 4)*x^2)); 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 ?

• A038208 → basis A013611 → (θ+1)²
• A085880 → basis A013611 → (θ+1)²
• A038221 → basis A013611 → (θ+1)²
• A085881 → basis A013611 → (θ+1)²
• A356546 → basis A013611 → (θ+1)²
• basis-map coefficients of A026729 → basis A013609 → (θ+1)²
• A156992 → basis A013611 → (θ+1)²
• A178820 → basis A013611 → (θ+1)²
• basis-map coefficients of A112227 → basis A013609 → (θ+1)²
• A013609 → basis A013610 → (θ+1)²
• A013609 → D → (θ+1)²
• basis-map coefficients of A030528, shifted → basis A013609 → (θ+1)²
• A120909 → basis A013610 → (θ+1)²
• A136572 → basis A013609 → (θ+1)²
• A130123 → basis A013609 → (θ+1)²
• A178822 → basis A013611 → (θ+1)²
• A120909 → (θ+1)²
• A013609 → (θ+1)²
• A196347 → basis A013611 → (θ+1)²
• A178821 → basis A013611 → (θ+1)²
• A028326 → basis A013611 → (θ+1)²

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.