A085881 → basis A007318 → (θ+1)²

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Submission text ?

Triangle begins:
  0:           1;
  1:           2,            4;
  2:          12,           48,            27;
  3:         120,          720,           810,           240;
  4:        1680,        13440,         22680,         13440,          2625;
  5:       30240,       302400,        680400,        604800,        236250,         34020;
  6:      665280,      7983360,      22453200,      26611200,      15592500,       4490640,        509355;
  7:    17297280,    242161920,     817296480,    1210809600,     945945000,     408648240,      92702610,      8648640;
  8:   518918400,   8302694400,   32691859200,   58118860800,   56756700000,   32691859200,   11124313200,   2075673600,   164189025;
  9: 17643225600, 317578060800, 1429101273600, 2964061900800, 3473510040000, 2500927228800, 1134679946400, 317578060800, 50241841650, 3445942500;

The row polynomials satisfy (4+(6*n-2)*x+n^2*x^2)*P(n,x) = (16*n-8+(24*n^2+12*n-12)*x+(4*n^3+18*n^2+2*n-6)*x^2+(2*n^3+3*n^2-1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 8 + (24*n^2 + 12*n - 12)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (2*n^3 + 3*n^2 - 1)*x^3)*P[n - 1, x])/(4 + (6*n - 2)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n - 8 + (24*n^2 + 12*n - 12)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (2*n^3 + 3*n^2 - 1)*x^3)*v[i - 1])/(4 + (6*n - 2)*x + 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
• 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_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • 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
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A196347 → x ↦ x+1 → (θ+1)²
• A178821 → basis A007318 → (θ+1)²
• A120910 → (θ+1)²
• A135065 → x ↦ x+1 → (θ+1)²
• A038325 → basis A007318 → (θ+1)²
• A038207 → (θ+1)²
• A085880 → x ↦ x+1 → (θ+1)²
• A178822 → x ↦ x+1 → (θ+1)²
• A028326 → basis A007318 → (θ+1)²
• A178820 → basis A007318 → (θ+1)²
• A120910 → D → (θ+1)²
• A038330 → basis A013609 → (θ+1)²
• A003506 → x ↦ x+1 → (θ+1)²
• A356546 → x ↦ x+1 → (θ+1)²
• A038208 → basis A007318 → (θ+1)²
• A156992 → basis A007318 → (θ+1)²
• A027466 → basis A013611 → (θ+1)²
• A038221 → x ↦ x+1 → (θ+1)²
• A038338 → x ↦ x+1 → (θ+1)²
• basis-map coefficients of A113214, shifted → x ↦ x+1 → (θ+1)²

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.