A069466 → basis A007318 → θ+1

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Submission text ?

Triangle begins:
  0:          1;
  1:          4,           4;
  2:         36,          72,          18;
  3:        400,        1200,         720,          80;
  4:       4900,       19600,       18900,        5600,         350;
  5:      63504,      317520,      423360,      211680,       37800,        1512;
  6:     853776,     5122656,     8731800,     6209280,     1940400,      232848,       6468;
  7:   11778624,    82450368,   171243072,   158558400,    72072000,    15567552,    1345344,     27456;
  8:  165636900,  1325095200,  3246483240,  3710266560,  2229727500,   713512800,  113513400,   7413120,   115830;
  9: 2363904400, 21275139600, 60070982400, 81763281600, 61322461200, 26281054800, 6289483200, 770140800, 39382200, 486200;

The row polynomials satisfy (2*n^2+n^3*x)*P(n,x) = (32*n^2-32*n+8+(16*n^3+32*n^2-52*n+16)*x+(12*n^3+2*n^2-16*n+6)*x^2)*P(n-1,x) - ((16*n-8)*x+(8*n^2+24*n-14)*x^2+(8*n^2+8*n-6)*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[((32*n^2 - 32*n + 8 + (16*n^3 + 32*n^2 - 52*n + 16)*x + (12*n^3 + 2*n^2 - 16*n + 6)*x^2)*P[n - 1, x] - ((16*n - 8)*x + (8*n^2 + 24*n - 14)*x^2 + (8*n^2 + 8*n - 6)*x^3)*D[P[n - 1, x], x])/(2*n^2 + n^3*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] = ((32*n^2 - 32*n + 8 + (16*n^3 + 32*n^2 - 52*n + 16)*x + (12*n^3 + 2*n^2 - 16*n + 6)*x^2)*v[i - 1] - ((16*n - 8)*x + (8*n^2 + 24*n - 14)*x^2 + (8*n^2 + 8*n - 6)*x^3)*deriv(v[i - 1], x))/(2*n^2 + n^3*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_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps a…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
        • 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
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Three-term recurrence criterion (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A008459 → x ↦ x+1 → θ+1
• A104684 → 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