A120909 → Veronese 2 (j=1) → (θ+1)²

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Submission text ?

Triangle begins:
  0:  0;
  1:  6;
  2: 12;
  3: 18,   96;
  4: 24,  384;
  5: 30,  960,    864;
  6: 36, 1920,   5184;
  7: 42, 3360,  18144,   6144;
  8: 48, 5376,  48384,  49152;
  9: 54, 8064, 108864, 221184, 38400;

The row polynomials satisfy (n-1-(10*n^3-28*n^2+22*n-4)*x+(4*n^5-12*n^4+12*n^3-4*n^2)*x^2)*P(n,x) = (n-(10*n^3-18*n^2-12*n+32)*x+(4*n^5-8*n^4-68*n^3+136*n^2+48*n-64)*x^2+(16*n^5-48*n^3-32*n^2)*x^3)*P(n-1,x) + (2*x-(20*n^2+4*n)*x^2+(8*n^4+16*n^3+88*n^2+16*n-32)*x^3-(32*n^4+64*n^3+32*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.

P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((n - (10*n^3 - 18*n^2 - 12*n + 32)*x + (4*n^5 - 8*n^4 - 68*n^3 + 136*n^2 + 48*n - 64)*x^2 + (16*n^5 - 48*n^3 - 32*n^2)*x^3)*P[n - 1, x] + (2*x - (20*n^2 + 4*n)*x^2 + (8*n^4 + 16*n^3 + 88*n^2 + 16*n - 32)*x^3 - (32*n^4 + 64*n^3 + 32*n^2)*x^4)*D[P[n - 1, x], x])/(n - 1 - (10*n^3 - 28*n^2 + 22*n - 4)*x + (4*n^5 - 12*n^4 + 12*n^3 - 4*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((n - (10*n^3 - 18*n^2 - 12*n + 32)*x + (4*n^5 - 8*n^4 - 68*n^3 + 136*n^2 + 48*n - 64)*x^2 + (16*n^5 - 48*n^3 - 32*n^2)*x^3)*v[i - 1] + (2*x - (20*n^2 + 4*n)*x^2 + (8*n^4 + 16*n^3 + 88*n^2 + 16*n - 32)*x^3 - (32*n^4 + 64*n^3 + 32*n^2)*x^4)*deriv(v[i - 1], x))/(n - 1 - (10*n^3 - 28*n^2 + 22*n - 4)*x + (4*n^5 - 12*n^4 + 12*n^3 - 4*n^2)*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 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_1 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., s…?
    • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal lengt…?
        • 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
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…?
            • 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
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal lengt…?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Iterated real-rootedness preservers (above)
                • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A229032 → (θ+1)²
• A013609 → Veronese 2 (j=1) → (θ+1)²

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.