A008459 → D → (θ+1)²

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Submission text ?

Triangle begins:
  0:  0;
  1:  1;
  2:  4,     8;
  3:  9,    72,     27;
  4: 16,   288,    432,      64;
  5: 25,   800,   2700,    1600,     125;
  6: 36,  1800,  10800,   14400,    4500,     216;
  7: 49,  3528,  33075,   78400,   55125,   10584,    343;
  8: 64,  6272,  84672,  313600,  392000,  169344,  21952,   512;
  9: 81, 10368, 190512, 1016064, 1984500, 1524096, 444528, 41472, 729;

The row polynomials satisfy -(n^3-3*n^2+3*n-1-(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x)*P(n,x) = -(n^3-n^2-(n^5-3*n^4-3*n^3+9*n^2)*x-(3*n^5-7*n^4+2*n^3+2*n^2)*x^2)*P(n-1,x) - (2*n^2*x-(2*n^4-n^3+n^2)*x^2+(2*n^4-n^3-n^2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - (n^5 - 3*n^4 - 3*n^3 + 9*n^2)*x - (3*n^5 - 7*n^4 + 2*n^3 + 2*n^2)*x^2)*P[n - 1, x] - (2*n^2*x - (2*n^4 - n^3 + n^2)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*D[P[n - 1, x], x])/(-n^3 + 3*n^2 - 3*n + 1 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - (n^5 - 3*n^4 - 3*n^3 + 9*n^2)*x - (3*n^5 - 7*n^4 + 2*n^3 + 2*n^2)*x^2)*v[i - 1] - (2*n^2*x - (2*n^4 - n^3 + n^2)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*deriv(v[i - 1], x))/(-n^3 + 3*n^2 - 3*n + 1 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 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 Square the entries of Pascal's triangle.?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Square the entries of Pascal's triangle.?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Square the entries of Pascal's triangle.?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Square the entries of Pascal's triangle.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Square the entries of Pascal's triangle.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A069466 → θ+1
• A008459 → 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
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.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023