A172431 → Veronese 2 → θ+1

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Submission text ?

Triangle begins:
  0: 1;
  1: 1;
  2: 1,   6;
  3: 1,  20;
  4: 1,  42,   15;
  5: 1,  72,  105;
  6: 1, 110,  378,   28;
  7: 1, 156,  990,  336;
  8: 1, 210, 2145, 1848,  45;
  9: 1, 272, 4095, 6864, 825;

The row polynomials satisfy P(n,x) = (12*x^2+x^3)*P(n-6,x) + 4*x^3*d/dx P(n-6,x) - (24*x+15*x^2)*P(n-5,x) - (12*x^2+4*x^3)*d/dx P(n-5,x) + (12+3*x-3*x^2)*P(n-4,x) + 12*x*d/dx P(n-4,x) + (27+10*x)*P(n-3,x) - (4+4*x-8*x^2)*d/dx P(n-3,x) - (11-3*x)*P(n-2,x) + (8+12*x)*d/dx P(n-2,x) + 5*P(n-1,x) - (4+4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 6*x, P(3,x) = 1 + 20*x, P(4,x) = 1 + 42*x + 15*x^2, P(5,x) = 1 + 72*x + 105*x^2.

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 6*x;
P[3, x_] := 1 + 20*x;
P[4, x_] := 1 + 42*x + 15*x^2;
P[5, x_] := 1 + 72*x + 105*x^2;
P[n_, x_] := P[n, x] = Expand[(12*x^2 + x^3)*P[n - 6, x] + 4*x^3*D[P[n - 6, x], x] - (24*x + 15*x^2)*P[n - 5, x] - (12*x^2 + 4*x^3)*D[P[n - 5, x], x] + (12 + 3*x - 3*x^2)*P[n - 4, x] + 12*x*D[P[n - 4, x], x] + (27 + 10*x)*P[n - 3, x] - (4 + 4*x - 8*x^2)*D[P[n - 3, x], x] - (11 - 3*x)*P[n - 2, x] + (8 + 12*x)*D[P[n - 2, x], x] + 5*P[n - 1, x] - (4 + 4*x)*D[P[n - 1, 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 + 6*x; v[4] = 1 + 20*x; v[5] = 1 + 42*x + 15*x^2; v[6] = 1 + 72*x + 105*x^2; for(n = 6, N, my(i = n + 1); v[i] = (12*x^2 + x^3)*v[i - 6] + 4*x^3*deriv(v[i - 6], x) - (24*x + 15*x^2)*v[i - 5] - (12*x^2 + 4*x^3)*deriv(v[i - 5], x) + (12 + 3*x - 3*x^2)*v[i - 4] + 12*x*deriv(v[i - 4], x) + (27 + 10*x)*v[i - 3] - (4 + 4*x - 8*x^2)*deriv(v[i - 3], x) - (11 - 3*x)*v[i - 2] + (8 + 12*x)*deriv(v[i - 2], x) + 5*v[i - 1] - (4 + 4*x)*deriv(v[i - 1], 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 veronese_section_2_0 of Even row Pascal-square read by antidiagonals.?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Even row Pascal-square read by antidiagonals.?
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Even row Pascal-square read by antidiagonals.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Even row Pascal-square read by antidiagonals.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Even row Pascal-square read by antidiagonals.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
4. [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
5. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
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.
8. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023