A360282 → basis A007318

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Triangle begins:
  0:      1;
  1:      1,       1;
  2:      5,       7,       2;
  3:     25,      43,      21,       3;
  4:    129,     261,     174,      46,       4;
  5:    681,    1581,    1320,     500,      85,      5;
  6:   3653,    9563,    9540,    4670,    1175,    141,     6;
  7:  19825,   57751,   66801,   39935,   13265,   2415,   217,    7;
  8: 108545,  348169,  457268,  322084,  132650,  32410,  4508,  316,   8;
  9: 598417, 2095513, 3076632, 2490768, 1222452, 374724, 70896, 7824, 441, 9;

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2+(4*n^2-10*n+4)*x+(2*n^2-5*n+2)*x^2)*P(n-2,x) + (12*n^2-24*n+10+(4*n^2-8*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2 + (4*n^2 - 10*n + 4)*x + (2*n^2 - 5*n + 2)*x^2)*P[n - 2, x] + (12*n^2 - 24*n + 10 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x])/(2*n^2 - 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2 + (4*n^2 - 10*n + 4)*x + (2*n^2 - 5*n + 2)*x^2)*v[i - 2] + (12*n^2 - 24*n + 10 + (4*n^2 - 8*n + 4)*x)*v[i - 1])/(2*n^2 - 3*n)); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, …?
    • 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
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, …?
    • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • ↑ Liu–Wang theorem for three-term recurrences (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. [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.