A062196 → D → (θ+1)²

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Submission text ?

Triangle begins:
  0:  0;
  1:  3;
  2:  8,    48;
  3: 15,   240,    270;
  4: 24,   720,   2160,     960;
  5: 35,  1680,   9450,   11200,    2625;
  6: 48,  3360,  30240,   67200,   42000,    6048;
  7: 63,  6048,  79380,  282240,  330750,  127008,   12348;
  8: 80, 10080, 181440,  940800, 1764000, 1270080,  329280,  23040;
  9: 99, 15840, 374220, 2661120, 7276500, 8382528, 4074840, 760320, 40095;

The row polynomials satisfy -(n^4-2*n^2+1-(n^6-3*n^4+3*n^2-1)*x)*P(n,x) = -(n^4+2*n^3-n^2-2*n-(n^6+2*n^5-8*n^4-12*n^3+7*n^2-2*n)*x-(3*n^6+8*n^5-n^4-4*n^3+10*n^2-4*n)*x^2)*P(n-1,x) - ((2*n^3+4*n^2)*x-(2*n^5+7*n^4+6*n^3+n^2+2*n)*x^2+(2*n^5+7*n^4+4*n^3-3*n^2+2*n)*x^3)*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 - (n^6 + 2*n^5 - 8*n^4 - 12*n^3 + 7*n^2 - 2*n)*x - (3*n^6 + 8*n^5 - n^4 - 4*n^3 + 10*n^2 - 4*n)*x^2)*P[n - 1, x] - ((2*n^3 + 4*n^2)*x - (2*n^5 + 7*n^4 + 6*n^3 + n^2 + 2*n)*x^2 + (2*n^5 + 7*n^4 + 4*n^3 - 3*n^2 + 2*n)*x^3)*D[P[n - 1, x], x])/(-n^4 + 2*n^2 - 1 + (n^6 - 3*n^4 + 3*n^2 - 1)*x)]];
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 - (n^6 + 2*n^5 - 8*n^4 - 12*n^3 + 7*n^2 - 2*n)*x - (3*n^6 + 8*n^5 - n^4 - 4*n^3 + 10*n^2 - 4*n)*x^2)*v[i - 1] - ((2*n^3 + 4*n^2)*x - (2*n^5 + 7*n^4 + 6*n^3 + n^2 + 2*n)*x^2 + (2*n^5 + 7*n^4 + 4*n^3 - 3*n^2 + 2*n)*x^3)*deriv(v[i - 1], x))/(-n^4 + 2*n^2 - 1 + (n^6 - 3*n^4 + 3*n^2 - 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 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 ?

• A108838 → reverse → (θ+1)²
• A062196 → θ+1

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.