A064861 → Veronese 2 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:     3,     2;
  1:     9,     8;
  2:    18,    21,     4;
  3:    54,    81,    28;
  4:   108,   189,    90,     8;
  5:   324,   675,   432,    80;
  6:   648,  1512,  1161,   312,   16;
  7:  1944,  5184,  4833,  1800,  208;
  8:  3888, 11340, 12015,  5490,  960,  32;
  9: 11664, 37908, 46413, 26136, 6480, 512;

The row polynomials satisfy P(n,x) = -(3*x+4*x^2)*P(n-4,x) + (6+4*x)*P(n-2,x) for n >= 4, with P(0,x) = 3 + 2*x, P(1,x) = 9 + 8*x, P(2,x) = 18 + 21*x + 4*x^2, P(3,x) = 54 + 81*x + 28*x^2.

P[0, x_] := 3 + 2*x;
P[1, x_] := 9 + 8*x;
P[2, x_] := 18 + 21*x + 4*x^2;
P[3, x_] := 54 + 81*x + 28*x^2;
P[n_, x_] := P[n, x] = Expand[-(3*x + 4*x^2)*P[n - 4, x] + (6 + 4*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; v[2] = 9 + 8*x; v[3] = 18 + 21*x + 4*x^2; v[4] = 54 + 81*x + 28*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(3*x + 4*x^2)*v[i - 4] + (6 + 4*x)*v[i - 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from veronese_section_2_0 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + …?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.?
        • 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 reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) f…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.?
                • ↑ Iterated real-rootedness preservers (above)
                • Parent-row provers of polytool polytool ? step › direct check
        • ↑ 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.