A086645 → Veronese 2 → θ+1

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Triangle begins:
   1: 1;
   2: 1;
   3: 1,    2;
   4: 1,   30;
   5: 1,  140,      3;
   6: 1,  420,    135;
   7: 1,  990,   1485,     4;
   8: 1, 2002,   9009,   364;
   9: 1, 3640,  38610,  7280,   5;
  10: 1, 6120, 131274, 74256, 765;

The row polynomials satisfy P(n,x) = (191-381*x+189*x^2+x^3)*P(n-6,x) - (64-192*x+192*x^2-64*x^3)*d/dx P(n-6,x) + (3342-2252*x-66*x^2)*P(n-5,x) - (128-1264*x+1120*x^2+16*x^3)*d/dx P(n-5,x) + (9041+146*x-3*x^2)*P(n-4,x) + (1152+5184*x+64*x^2)*d/dx P(n-4,x) + (1860+52*x)*P(n-3,x) - (2048-416*x-32*x^2)*d/dx P(n-3,x) - (111-3*x)*P(n-2,x) + (1472+128*x)*d/dx P(n-2,x) + 14*P(n-1,x) - (384+16*x)*d/dx P(n-1,x) for n >= 7, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 1 + 2*x, P(4,x) = 1 + 30*x, P(5,x) = 1 + 140*x + 3*x^2, P(6,x) = 1 + 420*x + 135*x^2.

P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 1 + 2*x;
P[4, x_] := 1 + 30*x;
P[5, x_] := 1 + 140*x + 3*x^2;
P[6, x_] := 1 + 420*x + 135*x^2;
P[n_, x_] := P[n, x] = Expand[(191 - 381*x + 189*x^2 + x^3)*P[n - 6, x] - (64 - 192*x + 192*x^2 - 64*x^3)*D[P[n - 6, x], x] + (3342 - 2252*x - 66*x^2)*P[n - 5, x] - (128 - 1264*x + 1120*x^2 + 16*x^3)*D[P[n - 5, x], x] + (9041 + 146*x - 3*x^2)*P[n - 4, x] + (1152 + 5184*x + 64*x^2)*D[P[n - 4, x], x] + (1860 + 52*x)*P[n - 3, x] - (2048 - 416*x - 32*x^2)*D[P[n - 3, x], x] - (111 - 3*x)*P[n - 2, x] + (1472 + 128*x)*D[P[n - 2, x], x] + 14*P[n - 1, x] - (384 + 16*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 1 + 2*x; v[4] = 1 + 30*x; v[5] = 1 + 140*x + 3*x^2; v[6] = 1 + 420*x + 135*x^2; for(n = 7, N, my(i = n); v[i] = (191 - 381*x + 189*x^2 + x^3)*v[i - 6] - (64 - 192*x + 192*x^2 - 64*x^3)*deriv(v[i - 6], x) + (3342 - 2252*x - 66*x^2)*v[i - 5] - (128 - 1264*x + 1120*x^2 + 16*x^3)*deriv(v[i - 5], x) + (9041 + 146*x - 3*x^2)*v[i - 4] + (1152 + 5184*x + 64*x^2)*deriv(v[i - 4], x) + (1860 + 52*x)*v[i - 3] - (2048 - 416*x - 32*x^2)*deriv(v[i - 3], x) - (111 - 3*x)*v[i - 2] + (1472 + 128*x)*deriv(v[i - 2], x) + 14*v[i - 1] - (384 + 16*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, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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) = binomial(2n, 2k).?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
        • 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 Triangle read by rows: T(n, k) = binomial(2n, 2k).?
            • ↑ Three-term recurrence criterion (above)
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (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