A137320 → Veronese 2 (j=1) → θ+1

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Triangle begins:
  0:     0;
  1:     2;
  2:     2;
  3:     4,      16;
  4:    12,      96;
  5:    48,     560,      96;
  6:   240,    3600,    1440;
  7:  1440,   25984,   16800,    512;
  8: 10080,  210112,  188160,  14336;
  9: 80640, 1889984, 2155104, 279552, 2560;

The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256-(8*n^2-64*n+128)*x+16*x^2)*P(n-4,x) + (4*n^3-42*n^2+148*n-175-(16*n-56)*x)*P(n-3,x) - (6*n^2-36*n+55-8*x)*P(n-2,x) + (4*n-10)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 2, P(3,x) = 4 + 16*x.

P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 2;
P[3, x_] := 4 + 16*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (8*n^2 - 64*n + 128)*x + 16*x^2)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 - (16*n - 56)*x)*P[n - 3, x] - (6*n^2 - 36*n + 55 - 8*x)*P[n - 2, x] + (4*n - 10)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 2; v[4] = 4 + 16*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (8*n^2 - 64*n + 128)*x + 16*x^2)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 - (16*n - 56)*x)*v[i - 3] - (6*n^2 - 36*n + 55 - 8*x)*v[i - 2] + (4*n - 10)*v[i - 1]); 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 Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n …?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
        • 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
        • Parent-row provers of polytool polytool ? step › direct check
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

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.