A091042 → Veronese 2 → D+1

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Triangle begins:
   1:    1;
   2:    1;
   3:    6,      5;
   4:   36,     35;
   5:  127,    144,      9;
   6:  331,    660,    165;
   7:  716,   3289,   1326,    13;
   8: 1366,  14235,   7800,   455;
   9: 2381,  51000,  42874,  6256,  17;
  10: 3877, 155040, 226746, 54264, 969;

The row polynomials satisfy P(n,x) = (127-189*x-3*x^2+65*x^3)*P(n-6,x) - (64-192*x+192*x^2-64*x^3)*d/dx P(n-6,x) + (3214-988*x-1186*x^2-16*x^3)*P(n-5,x) - (128-1264*x+1120*x^2+16*x^3)*d/dx P(n-5,x) + (10193+5330*x+61*x^2)*P(n-4,x) + (1152+5184*x+64*x^2)*d/dx P(n-4,x) - (188-468*x-32*x^2)*P(n-3,x) - (2048-416*x-32*x^2)*d/dx P(n-3,x) + (1361+131*x)*P(n-2,x) + (1472+128*x)*d/dx P(n-2,x) - (370+16*x)*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) = 6 + 5*x, P(4,x) = 36 + 35*x, P(5,x) = 127 + 144*x + 9*x^2, P(6,x) = 331 + 660*x + 165*x^2.

P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 6 + 5*x;
P[4, x_] := 36 + 35*x;
P[5, x_] := 127 + 144*x + 9*x^2;
P[6, x_] := 331 + 660*x + 165*x^2;
P[n_, x_] := P[n, x] = Expand[(127 - 189*x - 3*x^2 + 65*x^3)*P[n - 6, x] - (64 - 192*x + 192*x^2 - 64*x^3)*D[P[n - 6, x], x] + (3214 - 988*x - 1186*x^2 - 16*x^3)*P[n - 5, x] - (128 - 1264*x + 1120*x^2 + 16*x^3)*D[P[n - 5, x], x] + (10193 + 5330*x + 61*x^2)*P[n - 4, x] + (1152 + 5184*x + 64*x^2)*D[P[n - 4, x], x] - (188 - 468*x - 32*x^2)*P[n - 3, x] - (2048 - 416*x - 32*x^2)*D[P[n - 3, x], x] + (1361 + 131*x)*P[n - 2, x] + (1472 + 128*x)*D[P[n - 2, x], x] - (370 + 16*x)*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] = 6 + 5*x; v[4] = 36 + 35*x; v[5] = 127 + 144*x + 9*x^2; v[6] = 331 + 660*x + 165*x^2; for(n = 7, N, my(i = n); v[i] = (127 - 189*x - 3*x^2 + 65*x^3)*v[i - 6] - (64 - 192*x + 192*x^2 - 64*x^3)*deriv(v[i - 6], x) + (3214 - 988*x - 1186*x^2 - 16*x^3)*v[i - 5] - (128 - 1264*x + 1120*x^2 + 16*x^3)*deriv(v[i - 5], x) + (10193 + 5330*x + 61*x^2)*v[i - 4] + (1152 + 5184*x + 64*x^2)*deriv(v[i - 4], x) - (188 - 468*x - 32*x^2)*v[i - 3] - (2048 - 416*x - 32*x^2)*deriv(v[i - 3], x) + (1361 + 131*x)*v[i - 2] + (1472 + 128*x)*deriv(v[i - 2], x) - (370 + 16*x)*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
        • 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 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
                • ↑ 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. [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
4. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
5. [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
6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
7. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023