basis-map coefficients of A113214, shifted → Veronese 2 (j=1) → (θ+1)²

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Triangle begins:
  0:  0;
  1: -1;
  2:  2;
  3: -3,   -4;
  4:  4,   16;
  5: -5,  -40,    -9;
  6:  6,   80,    54;
  7: -7, -140,  -189,  -16;
  8:  8,  224,   504,  128;
  9: -9, -336, -1134, -576, -25;

The row polynomials satisfy (4*n-4-(10*n^3-28*n^2+22*n-4)*x+(n^5-3*n^4+3*n^3-n^2)*x^2)*P(n,x) = -(4*n-(10*n^3-18*n^2-12*n+32)*x+(n^5-2*n^4-17*n^3+34*n^2+12*n-16)*x^2+(n^5-3*n^3-2*n^2)*x^3)*P(n-1,x) - (8*x-(20*n^2+4*n)*x^2+(2*n^4+4*n^3+22*n^2+4*n-8)*x^3-(2*n^4+4*n^3+2*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = -1.

P[0, x_] := 0;
P[1, x_] := -1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - (10*n^3 - 18*n^2 - 12*n + 32)*x + (n^5 - 2*n^4 - 17*n^3 + 34*n^2 + 12*n - 16)*x^2 + (n^5 - 3*n^3 - 2*n^2)*x^3)*P[n - 1, x] - (8*x - (20*n^2 + 4*n)*x^2 + (2*n^4 + 4*n^3 + 22*n^2 + 4*n - 8)*x^3 - (2*n^4 + 4*n^3 + 2*n^2)*x^4)*D[P[n - 1, x], x])/(4*n - 4 - (10*n^3 - 28*n^2 + 22*n - 4)*x + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = -1; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - (10*n^3 - 18*n^2 - 12*n + 32)*x + (n^5 - 2*n^4 - 17*n^3 + 34*n^2 + 12*n - 16)*x^2 + (n^5 - 3*n^3 - 2*n^2)*x^3)*v[i - 1] - (8*x - (20*n^2 + 4*n)*x^2 + (2*n^4 + 4*n^3 + 22*n^2 + 4*n - 8)*x^3 - (2*n^4 + 4*n^3 + 2*n^2)*x^4)*deriv(v[i - 1], x))/(4*n - 4 - (10*n^3 - 28*n^2 + 22*n - 4)*x + (n^5 - 3*n^4 + 3*n^3 - n^2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 veronese_section_2_1 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
                • Parent-row provers of polytool polytool ? step › direct check
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Parent-row provers of polytool polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A178821 → Veronese 2 (j=1) → (θ+1)²
• A034867 → (θ+1)²
• A178822 → Veronese 2 (j=1) → (θ+1)²
• A038208 → Veronese 2 (j=1) → (θ+1)²
• A103406 → Veronese 2 (j=1) → θ+1
• A178820 → Veronese 2 (j=1) → (θ+1)²
• A028326 → Veronese 2 (j=1) → (θ+1)²
• A007318 → Veronese 2 (j=1) → (θ+1)²

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.