A008459 → basis A013609 → D+1

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Submission text ?

Triangle begins:
  0:      1;
  1:      4,       2;
  2:     18,      20,        4;
  3:     80,     156,       72,        8;
  4:    350,    1000,      840,      224,       16;
  5:   1512,    5740,     7280,     3600,      640,      32;
  6:   6468,   30744,    52920,    40320,    13440,    1728,      64;
  7:  27456,  157080,   343728,   361200,   188160,   45696,    4480,    128;
  8: 115830,  775632,  2066064,  2794176,  2032800,  779520,  145152,  11264,   256;
  9: 486200, 3732300, 11737440, 19507488, 18450432, 9979200, 2956800, 437760, 27648, 512;

The row polynomials satisfy (2*n^3-2*n^2+(4*n^2-5*n)*x+2*n*x^2)*P(n,x) = (8*n^3-4*n^2-8*n+4+(12*n^3+4*n^2-34*n+14)*x+(24*n^2-26*n+8)*x^2+(12*n-8)*x^3)*P(n-1,x) - ((4*n^2+2*n-2)*x+(8*n^2+12*n-6)*x^2+16*n*x^3+8*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^3 - 4*n^2 - 8*n + 4 + (12*n^3 + 4*n^2 - 34*n + 14)*x + (24*n^2 - 26*n + 8)*x^2 + (12*n - 8)*x^3)*P[n - 1, x] - ((4*n^2 + 2*n - 2)*x + (8*n^2 + 12*n - 6)*x^2 + 16*n*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 2*n^2 + (4*n^2 - 5*n)*x + 2*n*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^3 - 4*n^2 - 8*n + 4 + (12*n^3 + 4*n^2 - 34*n + 14)*x + (24*n^2 - 26*n + 8)*x^2 + (12*n - 8)*x^3)*v[i - 1] - ((4*n^2 + 2*n - 2)*x + (8*n^2 + 12*n - 6)*x^2 + 16*n*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 2*n^2 + (4*n^2 - 5*n)*x + 2*n*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 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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