A159854 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
  0:    3,     4,     1;
  1:    7,    13,     7,     1;
  2:   16,    38,    31,    10,     1;
  3:   36,   104,   113,    57,    13,     1;
  4:   80,   272,   368,   252,    91,    16,     1;
  5:  176,   688,  1112,   960,   475,   133,    19,     1;
  6:  384,  1696,  3184,  3312,  2080,   802,   183,    22,    1;
  7:  832,  4096,  8752, 10640,  8092,  3976,  1253,   241,   25,   1;
  8: 1792,  9728, 23296, 32384, 28896, 17248,  6944,  1848,  307,  28,  1;
  9: 3840, 22784, 60416, 94464, 96672, 67872, 33264, 11328, 2607, 381, 31, 1;

The row polynomials satisfy (n+5+(n+4)*x+x^2)*P(n,x) = (2*n+12+(3*n+16)*x+(n+7)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 4*x + x^2.

P[0, x_] := 3 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 12 + (3*n + 16)*x + (n + 7)*x^2 + x^3)*P[n - 1, x])/(n + 5 + (n + 4)*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3 + 4*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 12 + (3*n + 16)*x + (n + 7)*x^2 + x^3)*v[i - 1])/(n + 5 + (n + 4)*x + 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 shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
    • Affine, reversal and Möbius substitutions (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
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

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.