A143491 → basis A013609 → reverse

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:    4,    14,      12;
  1:    8,    48,      94,       60;
  2:   16,   144,     476,      684,       360;
  3:   32,   400,    1960,     4700,      5508,      2520;
  4:   64,  1056,    7120,    25080,     48616,     49104,      20160;
  5:  128,  2688,   23744,   114240,    322952,    535752,     482256,     181440;
  6:  256,  6656,   74368,   465920,   1788304,   4301024,    6322032,    5185440,     1814400;
  7:  512, 16128,  221952,  1749888,   8701728,  28273392,   59955328,   79913232,    60668640,    19958400;
  8: 1024, 38400,  637440,  6163200,  38402112, 160967520,  459191360,  879290400,  1080296064,   767940480,   239500800;
  9: 2048, 90112, 1774080, 20613120, 156925824, 821162496, 3010960480, 7728068480, 13591367328, 15579729792, 10462227840, 3113510400;

The row polynomials satisfy P(n,x) = (2+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 14*x + 12*x^2.

P[0, x_] := 4 + 14*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (n + 4)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 14*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + (n + 4)*x)*v[i - 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 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 Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013609 of Unsigned 2-Stirling numbers of the first kind.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A143491 → basis A013609
• A028338 → basis A013611 → x ↦ x+1
• A136124 → basis A013609 → reverse
• A136124 → basis A013609

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. [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.