A028326 → basis A007318 → θ+1

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Triangle begins:
  0:    2;
  1:    4,    4;
  2:    8,   16,     6;
  3:   16,   48,    36,     8;
  4:   32,  128,   144,    64,    10;
  5:   64,  320,   480,   320,   100,    12;
  6:  128,  768,  1440,  1280,   600,   144,   14;
  7:  256, 1792,  4032,  4480,  2800,  1008,  196,   16;
  8:  512, 4096, 10752, 14336, 11200,  5376, 1568,  256,  18;
  9: 1024, 9216, 27648, 43008, 40320, 24192, 9408, 2304, 324, 20;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + (2*n + 4)*x + (n + 1)*x^2)*v[i - 1])/(2 + n*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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by 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 shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • Affine, reversal and Möbius substitutions (elementary) 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)
    • Geometric basis map (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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

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

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.