A156992 → D+1

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Submission text ?

Triangle begins:
  0:        1;
  1:        4,         2;
  2:       18,        24,          6;
  3:       96,       216,        144,         24;
  4:      600,      1920,       2160,        960,        120;
  5:     4320,     18000,      28800,      21600,       7200,        720;
  6:    35280,    181440,     378000,     403200,     226800,      60480,       5040;
  7:   322560,   1975680,    5080320,    7056000,    5644800,    2540160,     564480,     40320;
  8:  3265920,  23224320,   71124480,  121927680,  127008000,   81285120,   30481920,   5806080,   362880;
  9: 36288000, 293932800, 1045094400, 2133734400, 2743372800, 2286144000, 1219276800, 391910400, 65318400, 3628800;

The row polynomials satisfy (n+x)*P(n,x) = (n^2+2*n+1+(n^2+3*n+2)*x+(n+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + 1 + (n^2 + 3*n + 2)*x + (n + 1)*x^2)*P[n - 1, x])/(n + x)]];
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] = ((n^2 + 2*n + 1 + (n^2 + 3*n + 2)*x + (n + 1)*x^2)*v[i - 1])/(n + 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
    • 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) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
            • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

              • from Triangle read by rows: row n consists of n zeros followed by n!.?
                • 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 polytool ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from Triangle read by rows: row n consists of n zeros followed by n!.?
                    • ↑ Iterated real-rootedness preservers (above)
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Parent-row provers of polytool (above)
                    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                      • from Triangle read by rows: row n consists of n zeros followed by n!.?
                        • ↑ Iterated real-rootedness preservers (above)
                        • ↑ Parent-row provers of polytool (above)
                        • ↑ Parent-row provers of polytool (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

              • from Triangle read by rows: row n consists of n zeros followed by n!.?
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
                • ↑ Laguerre’s theorem on multiplier sequences (above)
                • ↑ Laguerre’s theorem on multiplier sequences (above)
        • 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 polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
            • ↑ Laguerre’s theorem on multiplier sequences (above)
            • ↑ Laguerre’s theorem on multiplier sequences (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Iterated real-rootedness preservers (above)
    • ↑ Parent-row provers of polytool (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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