A085880 → basis A013609 → (θ+1)²

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Submission text ?

Triangle begins:
   1:       1;
   2:       2,        8;
   3:       8,       64,        72;
   4:      40,      480,      1080,        640;
   5:     224,     3584,     12096,      14336,       5600;
   6:    1344,    26880,    120960,     215040,     168000,       48384;
   7:    8448,   202752,   1140480,    2703360,    3168000,     1824768,      413952;
   8:   54912,  1537536,  10378368,   30750720,   48048000,    41513472,    18834816,    3514368;
   9:  366080, 11714560,  92252160,  328007680,  640640000,   738017280,   502261760,  187432960,   29652480;
  10: 2489344, 89616384, 806547456, 3345678336, 7841433600, 11291664384, 10246139904, 5735448576, 1814731776, 248934400;

The row polynomials satisfy (n+(3*n^2-4*n)*x+(n^3-2*n^2+n)*x^2)*P(n,x) = (8*n-12+(24*n^2-36*n)*x+(8*n^3+12*n^2-44*n+12)*x^2+(8*n^3-12*n^2)*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (24*n^2 - 36*n)*x + (8*n^3 + 12*n^2 - 44*n + 12)*x^2 + (8*n^3 - 12*n^2)*x^3)*P[n - 1, x])/(n + (3*n^2 - 4*n)*x + (n^3 - 2*n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (24*n^2 - 36*n)*x + (8*n^3 + 12*n^2 - 44*n + 12)*x^2 + (8*n^3 - 12*n^2)*x^3)*v[i - 1])/(n + (3*n^2 - 4*n)*x + (n^3 - 2*n^2 + 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  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 the n-th Catalan nu…?
    • 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 the n-th Catalan nu…?
        • Geometric basis map (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) [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)
        • Geometric basis map (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)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

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