A028326 → basis A013611 → θ+1

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Triangle begins:
  0:    2;
  1:    4,    16;
  2:    8,    64,     96;
  3:   16,   192,    576,     512;
  4:   32,   512,   2304,    4096,     2560;
  5:   64,  1280,   7680,   20480,    25600,    12288;
  6:  128,  3072,  23040,   81920,   153600,   147456,    57344;
  7:  256,  7168,  64512,  286720,   716800,  1032192,   802816,   262144;
  8:  512, 16384, 172032,  917504,  2867200,  5505024,  6422528,  4194304,  1179648;
  9: 1024, 36864, 442368, 2752512, 10321920, 24772608, 38535168, 37748736, 21233664, 5242880;

The row polynomials satisfy (1+2*n*x)*P(n,x) = (2+(4*n+8)*x+(8*n+8)*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[((2 + (4*n + 8)*x + (8*n + 8)*x^2)*P[n - 1, x])/(1 + 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] = ((2 + (4*n + 8)*x + (8*n + 8)*x^2)*v[i - 1])/(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 basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • Geometric basis map (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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
        • 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)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A156992 → basis A013611 → θ+1
• basis-map coefficients of A026729, shifted → basis A013609 → θ+1
• A120909 → basis A013610 → θ+1
• A356546 → basis A013611 → θ+1
• A136572 → basis A013609 → θ+1
• A013609 → θ+1
• basis-map coefficients of A030528, shifted → basis A013609 → θ+1
• A085881 → basis A013611 → θ+1
• basis-map coefficients of A026729 → basis A013609 → θ+1
• A013609 → basis A013610 → θ+1
• A116666 → x ↦ x/(x+1)
• A178822 → basis A013611 → θ+1
• A038208 → basis A013611 → θ+1
• A196347 → basis A013611 → θ+1
• A116666 → reverse → x ↦ x+1
• A038221 → basis A013611 → θ+1
• A178820 → basis A013611 → θ+1
• A130123 → basis A013609 → θ+1
• A178821 → basis A013611 → θ+1
• basis-map coefficients of A112227 → basis A013609 → θ+1
• A120909 → θ+1
• A134082 → basis A007318 → x ↦ x+1
• A013609 → D → θ+1
• A085880 → basis A013611 → 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.