A123519 → basis A013609 → D

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Triangle begins:
  0:       28,        32;
  1:      152,       352,        192;
  2:      744,      2592,       2880,       1024;
  3:     3436,     15968,      26880,      19456,       5120;
  4:    15284,     88768,     200448,     220160,     117760,      24576;
  5:    66224,    461376,    1307520,    1930240,    1566720,     663552,     114688;
  6:   281424,   2286528,    7797888,   14474240,   15800320,   10149888,    3555328,     524288;
  7:  1178196,  10936128,   43603200,   97555456,  134005760,  115777536,   61472768,   18350080,    2359296;
  8:  4874444,  50883552,  232231680,  608297984, 1008000000, 1096187904,  782630912,  353894400,   92012544,  10485760;
  9: 19973192, 231593632, 1190624832, 3574205440, 6939484160, 9107374080, 8184594432, 4974968832, 1958215680, 450887680, 46137344;

The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (4*n^2+19*n+21+(4*n^2+20*n+24)*x)*P(n-1,x) + (3*n+6+(8*n+16)*x+(4*n+8)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 28 + 32*x.

P[0, x_] := 28 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 19*n + 21 + (4*n^2 + 20*n + 24)*x)*P[n - 1, x] + (3*n + 6 + (8*n + 16)*x + (4*n + 8)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 28 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 19*n + 21 + (4*n^2 + 20*n + 24)*x)*v[i - 1] + (3*n + 6 + (8*n + 16)*x + (4*n + 8)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 5*n + 3)); 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 Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
                • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
                • ↑ Three-term recurrence criterion (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

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