Veronese parent of A109446 → Veronese 2 (j=1) → θ+1

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Triangle begins:
  0: 1;
  1: 1;
  2: 1,   2;
  3: 3,   2;
  4: 1,  12,   3;
  5: 5,  20,   3;
  6: 1,  30,  45,   4;
  7: 7,  70,  63,   4;
  8: 1,  56, 210, 112, 5;
  9: 9, 168, 378, 144, 5;

The row polynomials satisfy P(n,x) = -(13-27*x+15*x^2-x^3)*P(n-6,x) + (4-12*x+12*x^2-4*x^3)*d/dx P(n-6,x) - (23-10*x+3*x^2)*P(n-4,x) - (8-8*x^2)*d/dx P(n-4,x) + (5+3*x)*P(n-2,x) + (4-4*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 3 + 2*x, P(4,x) = 1 + 12*x + 3*x^2, P(5,x) = 5 + 20*x + 3*x^2.

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 3 + 2*x;
P[4, x_] := 1 + 12*x + 3*x^2;
P[5, x_] := 5 + 20*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(13 - 27*x + 15*x^2 - x^3)*P[n - 6, x] + (4 - 12*x + 12*x^2 - 4*x^3)*D[P[n - 6, x], x] - (23 - 10*x + 3*x^2)*P[n - 4, x] - (8 - 8*x^2)*D[P[n - 4, x], x] + (5 + 3*x)*P[n - 2, x] + (4 - 4*x)*D[P[n - 2, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1 + 2*x; v[4] = 3 + 2*x; v[5] = 1 + 12*x + 3*x^2; v[6] = 5 + 20*x + 3*x^2; for(n = 6, N, my(i = n + 1); v[i] = -(13 - 27*x + 15*x^2 - x^3)*v[i - 6] + (4 - 12*x + 12*x^2 - 4*x^3)*deriv(v[i - 6], x) - (23 - 10*x + 3*x^2)*v[i - 4] - (8 - 8*x^2)*deriv(v[i - 4], x) + (5 + 3*x)*v[i - 2] + (4 - 4*x)*deriv(v[i - 2], x)); 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 Binomial coefficients C(n,k) with n-k even, read by rows.?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.?
        • 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

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.