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

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Triangle begins:
   1:  1;
   2: -4, -4;
   3:  0, 18,   9;
   4:  0,  0, -48, -16;
   5:  0,  0,   0, 100,   25;
   6:  0,  0,   0,   0, -180, -36;
   7:  0,  0,   0,   0,    0, 294,   49;
   8:  0,  0,   0,   0,    0,   0, -448, -64;
   9:  0,  0,   0,   0,    0,   0,    0, 648,   81;
  10:  0,  0,   0,   0,    0,   0,    0,   0, -900, -100;

The row polynomials satisfy (n^3-4*n^2+5*n-2+(n^2-2*n+1)*x)*P(n,x) = -((n^3-n^2)*x+n^2*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[(-((n^3 - n^2)*x + n^2*x^2)*P[n - 1, x])/(n^3 - 4*n^2 + 5*n - 2 + (n^2 - 2*n + 1)*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] = (-((n^3 - n^2)*x + n^2*x^2)*v[i - 1])/(n^3 - 4*n^2 + 5*n - 2 + (n^2 - 2*n + 1)*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 theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= …?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= …?
        • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= …?
            • ↑ Laguerre’s theorem on multiplier sequences (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= …?
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• image under θ+1 → D+1
• A136572 → (θ+1)² → D+1
• basis-map coefficients of A026729 → D+1
• basis-map coefficients of A026729 → (θ+1)² → D+1
• basis-map coefficients of A030528, shifted → D → D+1
• A127773 → θ+1 → D+1
• A130123 → D → D+1
• A127648 → D → D+1
• image under θ+1 → (θ+1)² → D+1
• A130123 → θ+1 → D+1
• A131431 → D+1 → D
• A131431 → D+1 → reverse
• basis-map coefficients of A030528, shifted → (θ+1)² → D+1
• A130123 → D+1
• basis-map coefficients of A026729, shifted → D → D+1
• image under θ+1 → θ+1 → D+1
• A023531 → D+1
• A132440 → θ+1 → D+1
• basis-map coefficients of A026729, shifted → D+1
• A131436 → D+1
• A127773 → D+1
• basis-map coefficients of A026729 → D → D+1
• image under θ+1 → θ+1 → D+1 → reverse
• image under θ+1 → θ+1 → D+1 → D
• A131431 → (θ+1)² → D+1
• basis-map coefficients of A030528, shifted → D+1
• A131431 → θ+1 → D+1
• A136572 → θ+1 → D+1
• A127773 → D → D+1
• A136572 → D → D+1
• A130123 → (θ+1)² → D+1
• A127648 → (θ+1)² → D+1
• A131431 → D+1
• A132440 → D+1
• Hadamard-factor parent of A127773 → D+1
• basis-map coefficients of A026729, shifted → θ+1 → D+1
• basis-map coefficients of A030528, shifted → θ+1 → D+1
• A136572 → D+1
• basis-map coefficients of A021012 → θ+1 → D+1
• basis-map coefficients of A021012 → D+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.