A085880 → D → (θ+1)²

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Submission text ?

Triangle begins:
   1:     0;
   2:     1;
   3:     4,      16;
   4:    15,     120,      135;
   5:    56,     672,     1512,      896;
   6:   210,    3360,    11340,    13440,     5250;
   7:   792,   15840,    71280,   126720,    99000,    28512;
   8:  3003,   72072,   405405,   960960,  1126125,   648648,   147147;
   9: 11440,  320320,  2162160,  6406400, 10010000,  8648640,  3923920,   732160;
  10: 43758, 1400256, 11027016, 39207168, 76576500, 88216128, 60035976, 22404096, 3544398;

The row polynomials satisfy (n^2-2*n+(3*n^3-13*n^2+14*n)*x+(n^4-6*n^3+12*n^2-8*n)*x^2)*P(n,x) = (4*n^2-10*n+6+(12*n^3-42*n^2+48*n-18)*x+(4*n^4-6*n^3-16*n^2+36*n-18)*x^2+(4*n^4-18*n^3+30*n^2-22*n+6)*x^3)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 10*n + 6 + (12*n^3 - 42*n^2 + 48*n - 18)*x + (4*n^4 - 6*n^3 - 16*n^2 + 36*n - 18)*x^2 + (4*n^4 - 18*n^3 + 30*n^2 - 22*n + 6)*x^3)*P[n - 1, x])/(n^2 - 2*n + (3*n^3 - 13*n^2 + 14*n)*x + (n^4 - 6*n^3 + 12*n^2 - 8*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 10*n + 6 + (12*n^3 - 42*n^2 + 48*n - 18)*x + (4*n^4 - 6*n^3 - 16*n^2 + 36*n - 18)*x^2 + (4*n^4 - 18*n^3 + 30*n^2 - 22*n + 6)*x^3)*v[i - 1])/(n^2 - 2*n + (3*n^3 - 13*n^2 + 14*n)*x + (n^4 - 6*n^3 + 12*n^2 - 8*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A…?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
        • 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
        • 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 the n-th Catalan number (A000108).?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • 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 the n-th Catalan number (A000108).?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Iterated real-rootedness preservers (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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