A120910 → D → θ+1

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Triangle begins:
  0:    0;
  1:    3;
  2:   12,    12;
  3:   36,    72,     27;
  4:   96,   288,    216,     48;
  5:  240,   960,   1080,    480,     75;
  6:  576,  2880,   4320,   2880,    900,   108;
  7: 1344,  8064,  15120,  13440,   6300,  1512,   147;
  8: 3072, 21504,  48384,  53760,  33600, 12096,  2352,  192;
  9: 6912, 55296, 145152, 193536, 151200, 72576, 21168, 3456, 243;

The row polynomials satisfy (2*n-2+(n^2-2*n+1)*x)*P(n,x) = (4*n+(2*n^2+2*n)*x+n^2*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + (2*n^2 + 2*n)*x + n^2*x^2)*P[n - 1, x])/(2*n - 2 + (n^2 - 2*n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + (2*n^2 + 2*n)*x + n^2*x^2)*v[i - 1])/(2*n - 2 + (n^2 - 2*n + 1)*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 derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k …?
    • Laguerre’s theorem on differential operators [SP14] 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) [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) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Parent-row provers of polytool (above)
            • Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

• A038330 → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → x ↦ x+1 → θ+1
• A178820 → basis A007318 → θ+1
• A027466 → basis A013611 → θ+1
• A178822 → x ↦ x+1 → θ+1
• A356546 → x ↦ x+1 → θ+1
• A085881 → basis A007318 → θ+1
• A131431 → D+1 → basis A013609
• A196347 → x ↦ x+1 → θ+1
• A028326 → basis A007318 → θ+1
• A120910 → θ+1
• A038208 → basis A007318 → θ+1
• A038207 → θ+1
• image under θ+1 → θ+1 → D+1 → basis A013609
• A038338 → x ↦ x+1 → θ+1
• A085880 → x ↦ x+1 → θ+1
• A132440 → D+1 → basis A013609
• A135065 → x ↦ x+1 → θ+1
• A178821 → basis A007318 → θ+1
• A003506 → x ↦ x+1 → θ+1
• A156992 → basis A007318 → θ+1
• A038221 → x ↦ x+1 → θ+1
• A094305 → x ↦ x+1 → θ+1
• A038325 → basis A007318 → θ+1

References

1. [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
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.