A178820 → Veronese 2 → θ+1

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Triangle begins:
  0:   1;
  1:   4;
  2:  10,    20;
  3:  20,   120;
  4:  35,   420,   105;
  5:  56,  1120,   840;
  6:  84,  2520,  3780,   336;
  7: 120,  5040, 12600,  3360;
  8: 165,  9240, 34650, 18480,  825;
  9: 220, 15840, 83160, 73920, 9900;

The row polynomials satisfy -(6*n^2-6*n-(n^4-n^2)*x)*P(n,x) = -(6*n^2+12*n-18-(n^4+3*n^3-13*n^2-27*n+36)*x-(n^4+5*n^3+5*n^2-5*n-6)*x^2)*P(n-1,x) - ((12*n+36)*x-(2*n^3+12*n^2+34*n+48)*x^2+(2*n^3+12*n^2+22*n+12)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.

P[0, x_] := 1;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 + 12*n - 18 - (n^4 + 3*n^3 - 13*n^2 - 27*n + 36)*x - (n^4 + 5*n^3 + 5*n^2 - 5*n - 6)*x^2)*P[n - 1, x] - ((12*n + 36)*x - (2*n^3 + 12*n^2 + 34*n + 48)*x^2 + (2*n^3 + 12*n^2 + 22*n + 12)*x^3)*D[P[n - 1, x], x])/(-6*n^2 + 6*n + (n^4 - n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n^2 + 12*n - 18 - (n^4 + 3*n^3 - 13*n^2 - 27*n + 36)*x - (n^4 + 5*n^3 + 5*n^2 - 5*n - 6)*x^2)*v[i - 1] - ((12*n + 36)*x - (2*n^3 + 12*n^2 + 34*n + 48)*x^2 + (2*n^3 + 12*n^2 + 22*n + 12)*x^3)*deriv(v[i - 1], x))/(-6*n^2 + 6*n + (n^4 - n^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 veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A028326 → Veronese 2 → θ+1
• basis-map coefficients of A113214, shifted → Veronese 2 → θ+1
• A178821 → Veronese 2 → θ+1
• A178822 → Veronese 2 → θ+1
• A034839 → θ+1
• A038208 → Veronese 2 → θ+1
• A003506 → Veronese 2 → θ+1
• A228955 → θ+1
• A119462 → θ+1

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. [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.