A062196 → D → θ+1

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Submission text ?

Triangle begins:
  0:  0;
  1:  3;
  2:  8,   24;
  3: 15,  120,     90;
  4: 24,  360,    720,    240;
  5: 35,  840,   3150,   2800,     525;
  6: 48, 1680,  10080,  16800,    8400,    1008;
  7: 63, 3024,  26460,  70560,   66150,   21168,   1764;
  8: 80, 5040,  60480, 235200,  352800,  211680,  47040,  2880;
  9: 99, 7920, 124740, 665280, 1455300, 1397088, 582120, 95040, 4455;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 + 2*n^3 - n^2 - 2*n + (3*n^4 + 4*n^3 - 3*n^2 + 2*n)*x)*v[i - 1] + ((2*n^3 + 4*n^2)*x - (2*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^4 - 2*n^2 + 1)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from 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 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 ?

• A062196: Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
• A008459 → D → D
• A108838 → reverse → θ+1
• A062196 → reverse
• A281260 → 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.