A159854 → basis A013610 → D+1

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Submission text ?

Triangle begins:
  0:     7,     24,       9;
  1:    17,     87,     117,      27;
  2:    40,    270,     603,     486,       81;
  3:    92,    768,    2367,    3267,     1863,      243;
  4:   208,   2064,    8064,   15660,    15471,     6804,      729;
  5:   464,   5328,   25128,   62640,    87885,    67311,    24057,     2187;
  6:  1024,  13344,   73584,  223344,   405000,   442746,   276291,    83106,     6561;
  7:  2240,  32640,  205776,  734832,  1630692,  2313360,  2066715,  1086939,   282123,    19683;
  8:  4864,  78336,  555264, 2277504,  5969376, 10396512, 12083904,  9115416,  4139991,   944784,   59049;
  9: 10496, 185088, 1456128, 6739200, 20349792, 41967072, 60011280, 58996512, 38480265, 15372423, 3129597, 177147;

The row polynomials satisfy (3*n+11+(9*n+18)*x+9*x^2)*P(n,x) = (6*n+28+(27*n+96)*x+(27*n+99)*x^2+27*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 7 + 24*x + 9*x^2.

P[0, x_] := 7 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 28 + (27*n + 96)*x + (27*n + 99)*x^2 + 27*x^3)*P[n - 1, x])/(3*n + 11 + (9*n + 18)*x + 9*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 7 + 24*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 28 + (27*n + 96)*x + (27*n + 99)*x^2 + 27*x^3)*v[i - 1])/(3*n + 11 + (9*n + 18)*x + 9*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_A013610 of Riordan array (1-x,x/(1-x)).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.