A028326 → D → D+1

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Submission text ?

Triangle begins:
  0:   0;
  1:   2;
  2:   8,    4;
  3:  18,   24,    6;
  4:  32,   72,   48,    8;
  5:  50,  160,  180,   80,   10;
  6:  72,  300,  480,  360,  120,   12;
  7:  98,  504, 1050, 1120,  630,  168,   14;
  8: 128,  784, 2016, 2800, 2240, 1008,  224,  16;
  9: 162, 1152, 3528, 6048, 6300, 4032, 1512, 288, 18;

The row polynomials satisfy (n^2-2*n+1+(n-1)*x)*P(n,x) = (n^2+(n^2+n)*x+n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + (n^2 + n)*x + n*x^2)*P[n - 1, x])/(n^2 - 2*n + 1 + (n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + (n^2 + n)*x + n*x^2)*v[i - 1])/(n^2 - 2*n + 1 + (n - 1)*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • Laguerre’s theorem on differential operators [SP14] 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)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.?
        • 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
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.?
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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

Related ?

Same fingerprint ?

• basis-map coefficients of A026729 → basis A007318 → D+1
• A085880 → basis A013609 → D+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A131431 → x ↦ x+1 → θ+1
• A178821 → basis A013609 → θ+1
• A038325 → basis A013609 → θ+1
• A344565 → D+1
• A356546 → basis A013609 → D+1
• basis-map coefficients of A113214, shifted → D → D+1
• A038208 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A196347 → θ+1
• A038208 → θ+1
• A156992 → basis A013609 → D+1
• A376827 → θ+1
• A038325 → D+1
• A038208 → D → θ+1
• A156992 → θ+1
• A038221 → basis A013609 → D+1
• A094305 → D+1
• A003506 → D → θ+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• A038221 → θ+1
• A038338 → D+1
• A038338 → basis A013609 → D+1
• A093375 → D → D
• A038338 → basis A013609 → θ+1
• A124860 → θ+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A135838 → D+1
• A120910 → basis A013610 → D+1
• A196347 → D+1
• A027465 → basis A013611 → D+1
• A131431 → x ↦ x+1 → D+1
• A124860 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A178821 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A028326 → D+1
• A085881 → basis A013609 → θ+1
• A156992 → D+1
• A356546 → basis A013609 → θ+1
• A216973 → θ+1
• A156992 → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → θ+1
• A178822 → basis A013609 → θ+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A178820 → θ+1
• A093375 → D
• A027465 → basis A013611 → θ+1
• A376827 → D+1
• A038221 → D+1
• basis-map coefficients of A127647 → D+1
• A178820 → D+1
• A135065 → θ+1
• A178822 → θ+1
• basis-map coefficients of A127647 → θ+1
• A085880 → D → θ+1
• A178820 → basis A013609 → D+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• A085880 → D → D+1
• A196347 → basis A013609 → D+1
• A178822 → basis A013609 → D+1
• A085880 → basis A013609 → θ+1
• A134346 → θ+1
• A028326 → basis A013609 → θ+1
• A038208 → D → D+1
• A085881 → D+1
• A028326 → D → θ+1
• A356546 → θ+1
• A135838 → θ+1
• A120910 → basis A013610 → θ+1
• A178822 → D+1
• A196347 → basis A013609 → θ+1
• A356546 → D+1
• A178821 → D+1
• A038338 → θ+1
• A178820 → basis A013609 → θ+1
• A038330 → basis A013611 → D+1
• A038325 → basis A013609 → D+1
• A094305 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A085881 → θ+1
• A134346 → D+1
• A131431 → D+1 → basis A007318
• A085880 → D+1
• A135065 → D+1
• A085880 → θ+1
• A028326 → θ+1
• A178821 → basis A013609 → D+1
• A038221 → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → D+1
• A038330 → basis A013611 → θ+1
• A028326 → basis A013609 → D+1
• A038325 → θ+1
• A085881 → basis A013609 → D+1
• A344565 → 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.