A143491 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:          28,           32;
  1:         188,          384,          192;
  2:        1368,         3808,         3456,         1024;
  3:       11016,        37600,        47040,        25600,         5120;
  4:       98208,       388928,       601920,       455680,       168960,        24576;
  5:      964512,      4286016,      7750848,      7311360,      3799040,      1032192,       114688;
  6:    10370880,     50576256,    103224576,    114451456,     74547200,     28557312,      5963776,      524288;
  7:   121337280,    639305856,   1438927872,   1809497088,   1392276480,    671956992,    198868992,    33030144,    2359296;
  8:  1535880960,   8642368512,  21102969600,  29388247040,  25754803200,  14746411008,   5522227200,  1305477120,  176947200,  10485760;
  9: 20924455680, 124637838336, 326192815872, 494596382720, 481753676800, 315326398464, 140605538304, 42215669760, 8174960640, 922746880, 46137344;

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(8*n+24)*x+16*x^2)*P(n-2,x) + (2*n+7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 28 + 32*x, P(1,x) = 188 + 384*x + 192*x^2.

P[0, x_] := 28 + 32*x;
P[1, x_] := 188 + 384*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (8*n + 24)*x + 16*x^2)*P[n - 2, x] + (2*n + 7 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 28 + 32*x; v[2] = 188 + 384*x + 192*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (8*n + 24)*x + 16*x^2)*v[i - 2] + (2*n + 7 + 8*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 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 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 → 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. [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.