A085881 → basis A013609 → (θ+1)²

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Triangle begins:
  0:           1;
  1:           2,            8;
  2:          12,           96,           108;
  3:         120,         1440,          3240,           1920;
  4:        1680,        26880,         90720,         107520,          42000;
  5:       30240,       604800,       2721600,        4838400,        3780000,        1088640;
  6:      665280,     15966720,      89812800,      212889600,      249480000,      143700480,       32598720;
  7:    17297280,    484323840,    3269185920,     9686476800,    15135120000,    13076743680,     5932967040,     1107025920;
  8:   518918400,  16605388800,  130767436800,   464950886400,   908107200000,  1046139494400,   711956044800,   265686220800,    42032390400;
  9: 17643225600, 635156121600, 5716405094400, 23712495206400, 55576160640000, 80029671321600, 72619516569600, 40649991782400, 12861911462400, 1764322560000;

The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (4*n-2+(12*n^2+6*n-6)*x+(4*n^3+18*n^2+2*n-6)*x^2+(4*n^3+6*n^2-2)*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[((4*n - 2 + (12*n^2 + 6*n - 6)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (4*n^3 + 6*n^2 - 2)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*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] = ((4*n - 2 + (12*n^2 + 6*n - 6)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (4*n^3 + 6*n^2 - 2)*x^3)*v[i - 1])/(1 + (3*n - 1)*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_A013609 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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • ↑ Geometric basis map (elementary) (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)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A085880 → basis A013609 → (θ+1)²
• A038330 → basis A013611 → (θ+1)²
• A356546 → basis A013609 → (θ+1)²
• A178820 → basis A013609 → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• A344565 → (θ+1)²
• A125092 → reverse → D
• basis-map coefficients of A127647 → (θ+1)²
• A038221 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• A134346 → (θ+1)²
• A038325 → basis A013609 → (θ+1)²
• A038208 → D → (θ+1)²
• A085880 → D → (θ+1)²
• A028326 → (θ+1)²
• A356546 → (θ+1)²
• A135838 → (θ+1)²
• A156992 → (θ+1)²
• A085881 → (θ+1)²
• A124860 → (θ+1)²
• A028326 → D → (θ+1)²
• A178822 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
• A038208 → (θ+1)²
• A156992 → basis A013609 → (θ+1)²
• A178822 → basis A013609 → (θ+1)²
• A196347 → (θ+1)²
• A120910 → basis A013610 → (θ+1)²
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
• A125092 → reverse
• A178820 → (θ+1)²
• A038338 → (θ+1)²
• A376827 → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A038338 → basis A013609 → (θ+1)²
• A003506 → D → (θ+1)²
• A003506 → (θ+1)²
• A135065 → (θ+1)²
• A178821 → (θ+1)²
• basis-map coefficients of A113214, shifted → D → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• A196347 → basis A013609 → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• A038325 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
• A085880 → (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.