A272866 → Veronese 2 → D

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Triangle begins:
  0:   0;
  1:   1;
  2:  11,     2;
  3:  30,    60,      3;
  4:  58,   390,    174,       4;
  5:  95,  1370,   2055,     380,       5;
  6: 141,  3540,  11967,    7080,     705,      6;
  7: 196,  7602,  46599,   62132,   19005,   1176,     7;
  8: 260, 14420, 140784,  346060,  234640,  43260,  1820,    8;
  9: 333, 25020, 357462, 1425132, 1781415, 714924, 87570, 2664, 9;

The row polynomials satisfy (9*n^2-25*n+16-(13*n^2-25*n+12)*x+(4*n^2-10*n+6)*x^2)*P(n,x) = (9*n^2-7*n+(59*n^2-193*n)*x-(55*n^2-107*n)*x^2+(12*n^2-22*n)*x^3)*P(n-1,x) + (18*n*x-134*n*x^2+74*n*x^3-8*n*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[((9*n^2 - 7*n + (59*n^2 - 193*n)*x - (55*n^2 - 107*n)*x^2 + (12*n^2 - 22*n)*x^3)*P[n - 1, x] + (18*n*x - 134*n*x^2 + 74*n*x^3 - 8*n*x^4)*D[P[n - 1, x], x])/(9*n^2 - 25*n + 16 - (13*n^2 - 25*n + 12)*x + (4*n^2 - 10*n + 6)*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] = ((9*n^2 - 7*n + (59*n^2 - 193*n)*x - (55*n^2 - 107*n)*x^2 + (12*n^2 - 22*n)*x^3)*v[i - 1] + (18*n*x - 134*n*x^2 + 74*n*x^3 - 8*n*x^4)*deriv(v[i - 1], x))/(9*n^2 - 25*n + 16 - (13*n^2 - 25*n + 12)*x + (4*n^2 - 10*n + 6)*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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.