basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²

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Triangle begins:
   1:  1;
   2: -1,  -4;
   3:  1,   8,    9;
   4: -1, -12,  -27,   -16;
   5:  1,  16,   54,    64,    25;
   6: -1, -20,  -90,  -160,  -125,   -36;
   7:  1,  24,  135,   320,   375,   216,    49;
   8: -1, -28, -189,  -560,  -875,  -756,  -343,   -64;
   9:  1,  32,  252,   896,  1750,  2016,  1372,   512,   81;
  10: -1, -36, -324, -1344, -3150, -4536, -4116, -2304, -729, -100;

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

• basis-map coefficients of A113214, shifted → D → (θ+1)²
• basis-map coefficients of A127647 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• A178820 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A135065 → (θ+1)²
• A376827 → (θ+1)²
• A038221 → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• A125092 → reverse
• A085881 → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• A135838 → (θ+1)²
• A120910 → basis A013610 → (θ+1)²
• A028326 → (θ+1)²
• A038208 → D → (θ+1)²
• A124860 → (θ+1)²
• A196347 → (θ+1)²
• A038338 → basis A013609 → (θ+1)²
• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
• A178820 → basis A013609 → (θ+1)²
• A038325 → (θ+1)²
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A003506 → D → (θ+1)²
• A038330 → basis A013611 → (θ+1)²
• A028326 → D → (θ+1)²
• A085880 → D → (θ+1)²
• A038338 → (θ+1)²
• A038325 → basis A013609 → (θ+1)²
• A356546 → (θ+1)²
• A085880 → (θ+1)²
• A178821 → (θ+1)²
• A125092 → reverse → D
• A178822 → basis A013609 → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A085880 → basis A013609 → (θ+1)²
• A156992 → basis A013609 → (θ+1)²
• A344565 → (θ+1)²
• A134346 → (θ+1)²
• A156992 → (θ+1)²
• A196347 → basis A013609 → (θ+1)²
• A003506 → (θ+1)²
• A178822 → (θ+1)²
• A038208 → (θ+1)²
• A085881 → basis A013609 → (θ+1)²
• A356546 → basis A013609 → (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.