A062196 → D → D

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Submission text ?

Triangle begins:
  0:    0;
  1:    0;
  2:   12;
  3:   60,    60;
  4:  180,   480,    180;
  5:  420,  2100,   2100,     420;
  6:  840,  6720,  12600,    6720,     840;
  7: 1512, 17640,  52920,   52920,   17640,   1512;
  8: 2520, 40320, 176400,  282240,  176400,  40320,  2520;
  9: 3960, 83160, 498960, 1164240, 1164240, 498960, 83160, 3960;

The row polynomials satisfy (n-2)*P(n,x) = (n+2+(3*n-4)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 12.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 12;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (3*n - 4)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 12; for(n = 3, N, my(i = n + 1); v[i] = ((n + 2 + (3*n - 4)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x))/(n - 2)); 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 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, 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 reversal of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)…
• A108838 → reverse → D

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.