A159854 → Veronese 2 (j=1) → D+1

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Triangle begins:
  0: 0;
  1: 1,   1;
  2: 2,   2;
  3: 3,   5,   1;
  4: 4,  12,   4;
  5: 5,  25,  13,   1;
  6: 6,  46,  38,   6;
  7: 7,  77,  98,  25,  1;
  8: 8, 120, 224,  88,  8;
  9: 9, 177, 462, 270, 41, 1;

The row polynomials satisfy -(2*n-2-(n^3-2*n^2-n+2)*x-(4*n^2-14*n+10)*x^2-(4*n-4)*x^3)*P(n,x) = -(2*n-(n^3-n^2-6*n+12)*x-(n^3+3*n^2-26*n+20)*x^2-(4*n^2-2*n-8)*x^3-4*n*x^4)*P(n-1,x) - (4*x-(2*n^2+2*n-8)*x^2+(2*n^2-6*n-8)*x^3+(8*n-12)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1 + x.

P[0, x_] := 0;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - (n^3 - n^2 - 6*n + 12)*x - (n^3 + 3*n^2 - 26*n + 20)*x^2 - (4*n^2 - 2*n - 8)*x^3 - 4*n*x^4)*P[n - 1, x] - (4*x - (2*n^2 + 2*n - 8)*x^2 + (2*n^2 - 6*n - 8)*x^3 + (8*n - 12)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-2*n + 2 + (n^3 - 2*n^2 - n + 2)*x + (4*n^2 - 14*n + 10)*x^2 + (4*n - 4)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - (n^3 - n^2 - 6*n + 12)*x - (n^3 + 3*n^2 - 26*n + 20)*x^2 - (4*n^2 - 2*n - 8)*x^3 - 4*n*x^4)*v[i - 1] - (4*x - (2*n^2 + 2*n - 8)*x^2 + (2*n^2 - 6*n - 8)*x^3 + (8*n - 12)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-2*n + 2 + (n^3 - 2*n^2 - n + 2)*x + (4*n^2 - 14*n + 10)*x^2 + (4*n - 4)*x^3)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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_1 of Riordan array (1-x,x/(1-x)).?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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.