A391533 → Veronese 2 (j=1) → θ+1

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Triangle begins:
  0:  4;
  1:  8,   2;
  2: 12,  12;
  3: 16,  38,    3;
  4: 20,  88,   24;
  5: 24, 170,  102,    4;
  6: 28, 292,  312,   40;
  7: 32, 462,  777,  212,   5;
  8: 36, 688, 1680,  800,  60;
  9: 40, 978, 3276, 2424, 380, 6;

The row polynomials satisfy -(32*n-(16*n^3+16*n^2+8*n+48)*x+(8*n^3+28*n^2+32*n+12)*x^2-(n^3+6*n^2+11*n+6)*x^3)*P(n,x) = -(32*n+32-(16*n^3+32*n^2-40*n+152)*x-(8*n^3-12*n^2-88*n-24)*x^2+(7*n^3+19*n^2-10*n-24)*x^3-(n^3+5*n^2+6*n)*x^4)*P(n-1,x) - (64*x-(32*n^2+96*n+112)*x^2+(48*n^2+160*n+104)*x^3-(18*n^2+74*n+68)*x^4+(2*n^2+10*n+12)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4.

P[0, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(32*n + 32 - (16*n^3 + 32*n^2 - 40*n + 152)*x - (8*n^3 - 12*n^2 - 88*n - 24)*x^2 + (7*n^3 + 19*n^2 - 10*n - 24)*x^3 - (n^3 + 5*n^2 + 6*n)*x^4)*P[n - 1, x] - (64*x - (32*n^2 + 96*n + 112)*x^2 + (48*n^2 + 160*n + 104)*x^3 - (18*n^2 + 74*n + 68)*x^4 + (2*n^2 + 10*n + 12)*x^5)*D[P[n - 1, x], x])/(-32*n + (16*n^3 + 16*n^2 + 8*n + 48)*x - (8*n^3 + 28*n^2 + 32*n + 12)*x^2 + (n^3 + 6*n^2 + 11*n + 6)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 4; for(n = 1, N, my(i = n + 1); v[i] = (-(32*n + 32 - (16*n^3 + 32*n^2 - 40*n + 152)*x - (8*n^3 - 12*n^2 - 88*n - 24)*x^2 + (7*n^3 + 19*n^2 - 10*n - 24)*x^3 - (n^3 + 5*n^2 + 6*n)*x^4)*v[i - 1] - (64*x - (32*n^2 + 96*n + 112)*x^2 + (48*n^2 + 160*n + 104)*x^3 - (18*n^2 + 74*n + 68)*x^4 + (2*n^2 + 10*n + 12)*x^5)*deriv(v[i - 1], x))/(-32*n + (16*n^3 + 16*n^2 + 8*n + 48)*x - (8*n^3 + 28*n^2 + 32*n + 12)*x^2 + (n^3 + 6*n^2 + 11*n + 6)*x^3)); 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 The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n…?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
                • ↑ Liu–Wang theorem (above)
                • 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
        • ↑ 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.