A143491 → basis A007318 → reverse

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Submission text ?

Triangle begins:
  0: 1,  7,   12;
  1: 1, 12,   47,    60;
  2: 1, 18,  119,   342,     360;
  3: 1, 25,  245,  1175,    2754,     2520;
  4: 1, 33,  445,  3135,   12154,    24552,    20160;
  5: 1, 42,  742,  7140,   40369,   133938,   241128,    181440;
  6: 1, 52, 1162, 14560,  111769,   537628,  1580508,   2592720,    1814400;
  7: 1, 63, 1734, 27342,  271929,  1767087,  7494416,  19978308,   30334320,   19958400;
  8: 1, 75, 2490, 48150,  600033,  5030235, 28699460, 109911300,  270074016,  383970240,  239500800;
  9: 1, 88, 3465, 80520, 1225983, 12830664, 94092515, 483004280, 1698920916, 3894932448, 5231113920, 3113510400;

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

P[0, x_] := 1 + 7*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (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] = 1 + 7*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (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_A007318 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)
    • Affine, reversal and Möbius substitutions (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 → x ↦ x+1
• A143491 → basis A007318
• A136124 → x ↦ x+1 → reverse
• A145324 → x ↦ x/(x+1)

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.