A008459 → basis A013611 → θ+1

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Submission text ?

Triangle begins:
  0:     1;
  1:     2,       8;
  2:     6,      48,       48;
  3:    20,     240,      576,       256;
  4:    70,    1120,     4320,      5120,      1280;
  5:   252,    5040,    26880,     53760,     38400,      6144;
  6:   924,   22176,   151200,    430080,    537600,    258048,     28672;
  7:  3432,   96096,   798336,   2956800,   5376000,   4644864,   1605632,    131072;
  8: 12870,  411840,  4036032,  18450432,  44352000,  56770560,  36126720,   9437184,   589824;
  9: 48620, 1750320, 19768320, 107627520, 322882560, 553512960, 529858560, 259522560, 53084160, 2621440;

The row polynomials satisfy (n+2*n^2*x)*P(n,x) = (4*n-2+(8*n^2+20*n-16)*x+(24*n^2+16*n-24)*x^2)*P(n-1,x) - (2*x+(4*n+14)*x^2+(16*n+24)*x^3)*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[((4*n - 2 + (8*n^2 + 20*n - 16)*x + (24*n^2 + 16*n - 24)*x^2)*P[n - 1, x] - (2*x + (4*n + 14)*x^2 + (16*n + 24)*x^3)*D[P[n - 1, x], x])/(n + 2*n^2*x)]];
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] = ((4*n - 2 + (8*n^2 + 20*n - 16)*x + (24*n^2 + 16*n - 24)*x^2)*v[i - 1] - (2*x + (4*n + 14)*x^2 + (16*n + 24)*x^3)*deriv(v[i - 1], x))/(n + 2*n^2*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
• 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_A013611 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