basis-map coefficients of A297191, shifted → D → basis A013610

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Submission text ?

Triangle begins:
  0:  0;
  1:  2,   6;
  2:  4,  36,   108,    108;
  3:  6,  90,   540,   1620,    2430,     1458;
  4:  8, 168,  1512,   7560,   22680,    40824,     40824,     17496;
  5: 10, 270,  3240,  22680,  102060,   306180,    612360,    787320,     590490,     196830;
  6: 12, 396,  5940,  53460,  320760,  1347192,   4041576,   8660520,   12990780,   12990780,     7794468,     2125764;
  7: 14, 546,  9828, 108108,  810810,  4378374,  17513496,  52540488,  118216098,  197026830,   236432196,   193444524,    96722262,    22320522;
  8: 16, 720, 15120, 196560, 1769040, 11675664,  58378320, 225173520,  675520560, 1576214640,  2837186352,  3868890480,  3868890480,  2678462640,  1147912560,   229582512;
  9: 18, 918, 22032, 330480, 3470040, 27066312, 162397872, 765589968, 2870962380, 8612887140, 20670929136, 39462682896, 59194024344, 68300797320, 58543540560, 35126124336, 13172296626, 2324522934;

The row polynomials satisfy (n-1)*P(n,x) = (n+6*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 6*x.

P[0, x_] := 0;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 6*n*x + 9*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 6*n*x + 9*n*x^2)*v[i - 1])/(n - 1)); 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delanno…?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A297191, shifted → basis A013610 → D

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.