A028326 → Veronese 2 (j=1) → D+1

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Triangle begins:
  0:   0;
  1:   2;
  2:   4;
  3:   8,   2;
  4:  16,   8;
  5:  30,  24,   2;
  6:  52,  64,  12;
  7:  84, 154,  48,  2;
  8: 128, 336, 160, 16;
  9: 186, 672, 468, 80, 2;

The row polynomials satisfy (n^3-6*n^2+17*n-12+(4*n^2-22*n+18)*x+(4*n-4)*x^2)*P(n,x) = (n^3-3*n^2+8*n+(n^3-3*n^2-2*n-12)*x+(4*n^2-18*n+28)*x^2+(4*n-8)*x^3)*P(n-1,x) + ((2*n^2-6*n+8)*x-(2*n^2-14*n+28)*x^2-(8*n-28)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n^2 + 8*n + (n^3 - 3*n^2 - 2*n - 12)*x + (4*n^2 - 18*n + 28)*x^2 + (4*n - 8)*x^3)*P[n - 1, x] + ((2*n^2 - 6*n + 8)*x - (2*n^2 - 14*n + 28)*x^2 - (8*n - 28)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^3 - 6*n^2 + 17*n - 12 + (4*n^2 - 22*n + 18)*x + (4*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - 3*n^2 + 8*n + (n^3 - 3*n^2 - 2*n - 12)*x + (4*n^2 - 18*n + 28)*x^2 + (4*n - 8)*x^3)*v[i - 1] + ((2*n^2 - 6*n + 8)*x - (2*n^2 - 14*n + 28)*x^2 - (8*n - 28)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^3 - 6*n^2 + 17*n - 12 + (4*n^2 - 22*n + 18)*x + (4*n - 4)*x^2)); 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 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
        • 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 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • 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
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A007318 → Veronese 2 (j=1) → D+1
• A178821 → Veronese 2 (j=1) → D+1
• A038208 → Veronese 2 (j=1) → D+1
• A178820 → Veronese 2 (j=1) → D+1
• A178822 → Veronese 2 (j=1) → D+1
• A003506 → Veronese 2 (j=1) → D+1
• basis-map coefficients of A113214, shifted → Veronese 2 (j=1) → D+1
• A034867 → D+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.