A008459 → basis A013609 → (θ+1)²

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Triangle begins:
  0:     1;
  1:     2,       8;
  2:     6,      48,       36;
  3:    20,     240,      432,      128;
  4:    70,    1120,     3240,     2560,       400;
  5:   252,    5040,    20160,    26880,     12000,      1152;
  6:   924,   22176,   113400,   215040,    168000,     48384,     3136;
  7:  3432,   96096,   598752,  1478400,   1680000,    870912,   175616,     8192;
  8: 12870,  411840,  3027024,  9225216,  13860000,  10644480,  3951360,   589824,   20736;
  9: 48620, 1750320, 14826240, 53813760, 100900800, 103783680, 57953280, 16220160, 1866240, 51200;

The row polynomials satisfy (2*n^2+(6*n^3-3*n^2)*x+2*n^4*x^2)*P(n,x) = (8*n^2-4*n+(24*n^3+24*n^2-26*n)*x+(8*n^4+56*n^3+4*n^2-44*n+4)*x^2+(12*n^4+24*n^3-4*n^2-16*n)*x^3)*P(n-1,x) - (4*n*x+(12*n^2+20*n-1)*x^2+(4*n^3+36*n^2+32*n-2)*x^3+(8*n^3+24*n^2+16*n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 4*n + (24*n^3 + 24*n^2 - 26*n)*x + (8*n^4 + 56*n^3 + 4*n^2 - 44*n + 4)*x^2 + (12*n^4 + 24*n^3 - 4*n^2 - 16*n)*x^3)*P[n - 1, x] - (4*n*x + (12*n^2 + 20*n - 1)*x^2 + (4*n^3 + 36*n^2 + 32*n - 2)*x^3 + (8*n^3 + 24*n^2 + 16*n)*x^4)*D[P[n - 1, x], x])/(2*n^2 + (6*n^3 - 3*n^2)*x + 2*n^4*x^2)]];
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] = ((8*n^2 - 4*n + (24*n^3 + 24*n^2 - 26*n)*x + (8*n^4 + 56*n^3 + 4*n^2 - 44*n + 4)*x^2 + (12*n^4 + 24*n^3 - 4*n^2 - 16*n)*x^3)*v[i - 1] - (4*n*x + (12*n^2 + 20*n - 1)*x^2 + (4*n^3 + 36*n^2 + 32*n - 2)*x^3 + (8*n^3 + 24*n^2 + 16*n)*x^4)*deriv(v[i - 1], x))/(2*n^2 + (6*n^3 - 3*n^2)*x + 2*n^4*x^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 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by 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 basis_map_A013609 of Square the entries of Pascal's triangle.?
    • Geometric basis map (elementary) 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)
    • Geometric basis map (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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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