A272866 → Veronese 2 → θ+1

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Triangle begins:
  0: 1;
  1: 1,   2;
  2: 1,  22,     3;
  3: 1,  60,    90,      4;
  4: 1, 116,   585,    232,       5;
  5: 1, 190,  2055,   2740,     475,       6;
  6: 1, 282,  5310,  15956,    8850,     846,      7;
  7: 1, 392, 11403,  62132,   77665,   22806,   1372,      8;
  8: 1, 520, 21630, 187712,  432575,  281568,  50470,   2080,    9;
  9: 1, 666, 37530, 476616, 1781415, 2137698, 834078, 100080, 2997, 10;

The row polynomials satisfy -(6*n-6-(9*n^3-6*n^2+9*n-12)*x+(13*n^3-2*n^2+5*n-16)*x^2-(4*n^3-6*n^2+2*n)*x^3)*P(n,x) = -(6*n-6-(9*n^3-6*n^2-87*n+84)*x-(59*n^3+70*n^2+21*n-150)*x^2+(55*n^3+52*n^2-23*n-84)*x^3-(12*n^3-2*n^2-16*n+6)*x^4)*P(n-1,x) - (12*x-(18*n^2+38*n+112)*x^2+(134*n^2+274*n+204)*x^3-(74*n^2+94*n)*x^4+(8*n^2+8*n-4)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 6 - (9*n^3 - 6*n^2 - 87*n + 84)*x - (59*n^3 + 70*n^2 + 21*n - 150)*x^2 + (55*n^3 + 52*n^2 - 23*n - 84)*x^3 - (12*n^3 - 2*n^2 - 16*n + 6)*x^4)*P[n - 1, x] - (12*x - (18*n^2 + 38*n + 112)*x^2 + (134*n^2 + 274*n + 204)*x^3 - (74*n^2 + 94*n)*x^4 + (8*n^2 + 8*n - 4)*x^5)*D[P[n - 1, x], x])/(-6*n + 6 + (9*n^3 - 6*n^2 + 9*n - 12)*x - (13*n^3 - 2*n^2 + 5*n - 16)*x^2 + (4*n^3 - 6*n^2 + 2*n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 6 - (9*n^3 - 6*n^2 - 87*n + 84)*x - (59*n^3 + 70*n^2 + 21*n - 150)*x^2 + (55*n^3 + 52*n^2 - 23*n - 84)*x^3 - (12*n^3 - 2*n^2 - 16*n + 6)*x^4)*v[i - 1] - (12*x - (18*n^2 + 38*n + 112)*x^2 + (134*n^2 + 274*n + 204)*x^3 - (74*n^2 + 94*n)*x^4 + (8*n^2 + 8*n - 4)*x^5)*deriv(v[i - 1], x))/(-6*n + 6 + (9*n^3 - 6*n^2 + 9*n - 12)*x - (13*n^3 - 2*n^2 + 5*n - 16)*x^2 + (4*n^3 - 6*n^2 + 2*n)*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_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
            • 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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
        • ↑ Veronese sections of Pólya frequency sequences (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.