A085880 → D → D+1

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Submission text ?

Triangle begins:
   1:      0;
   2:      1;
   3:      8,       4;
   4:     45,      60,      15;
   5:    224,     504,     336,       56;
   6:   1050,    3360,    3780,     1680,      210;
   7:   4752,   19800,   31680,    23760,     7920,     792;
   8:  21021,  108108,  225225,   240240,   135135,   36036,    3003;
   9:  91520,  560560, 1441440,  2002000,  1601600,  720720,  160160,  11440;
  10: 393822, 2800512, 8576568, 14702688, 15315300, 9801792, 3675672, 700128, 43758;

The row polynomials satisfy (n^3-4*n^2+4*n+(n^2-2*n)*x)*P(n,x) = (4*n^3-14*n^2+16*n-6+(4*n^3-10*n^2+6*n)*x+(4*n^2-10*n+6)*x^2)*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^3 - 14*n^2 + 16*n - 6 + (4*n^3 - 10*n^2 + 6*n)*x + (4*n^2 - 10*n + 6)*x^2)*P[n - 1, x])/(n^3 - 4*n^2 + 4*n + (n^2 - 2*n)*x)]];
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^3 - 14*n^2 + 16*n - 6 + (4*n^3 - 10*n^2 + 6*n)*x + (4*n^2 - 10*n + 6)*x^2)*v[i - 1])/(n^3 - 4*n^2 + 4*n + (n^2 - 2*n)*x)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 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 ?

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