A143491 → basis A013611 → reverse

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Submission text ?

Triangle begins:
  0:      16,       28,        12;
  1:      64,      192,       188,         60;
  2:     256,     1152,      1904,       1368,         360;
  3:    1024,     6400,     15680,      18800,       11016,        2520;
  4:    4096,    33792,    113920,     200640,      194464,       98208,       20160;
  5:   16384,   172032,    759808,    1827840,     2583616,     2143008,      964512,       181440;
  6:   65536,   851968,   4759552,   14909440,    28612864,    34408192,    25288128,     10370880,      1814400;
  7:  262144,  4128768,  28409856,  111992832,   278455296,   452374272,   479642624,    319652928,    121337280,    19958400;
  8: 1048576, 19660800, 163184640,  788889600,  2457735168,  5150960640,  7347061760,   7034323200,   4321184256,  1535880960,   239500800;
  9: 4194304, 92274688, 908328960, 5276958720, 20086505472, 52554399744, 96350735360, 123649095680, 108730938624, 62318919168, 20924455680, 3113510400;

The row polynomials satisfy P(n,x) = (4+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + 28*x + 12*x^2.

P[0, x_] := 16 + 28*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + (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] = 16 + 28*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + (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_A013611 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 ?

• A136124 → basis A013611 → reverse
• A028338 → basis A013611 → basis A013609
• A143491 → basis A013611
• A136124 → basis A013611

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.