A143491 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
  0:         12,          42,           27;
  1:         60,         282,          324,          108;
  2:        360,        2052,         3213,         1944,          405;
  3:       2520,       16524,        31725,        26460,        10125,         1458;
  4:      20160,      147312,       328158,       338580,       180225,        48114,        5103;
  5:     181440,     1446768,      3616326,      4359852,      2891700,      1081836,      214326,       17496;
  6:    1814400,    15556320,     42673716,     58063824,     45266445,     21228480,     5929686,      909792,      59049;
  7:   19958400,   182005920,    539414316,    809396928,    715670235,    396472482,   139526226,    30338064,    3720087,    196830;
  8:  239500800,  2303821440,   7291998432,  11870420400,  11623281300,   7334082630,  3061968399,   842432400,  147032010,  14762250,   649539;
  9: 3113510400, 31386683520, 105163176096, 183483458928, 195616733400, 137186886870, 65474878392, 21449798568, 4754625480, 682015950, 57159432, 2125764;

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(6*n+18)*x+9*x^2)*P(n-2,x) + (2*n+7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 12 + 42*x + 27*x^2, P(1,x) = 60 + 282*x + 324*x^2 + 108*x^3.

P[0, x_] := 12 + 42*x + 27*x^2;
P[1, x_] := 60 + 282*x + 324*x^2 + 108*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (6*n + 18)*x + 9*x^2)*P[n - 2, x] + (2*n + 7 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 42*x + 27*x^2; v[2] = 60 + 282*x + 324*x^2 + 108*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (6*n + 18)*x + 9*x^2)*v[i - 2] + (2*n + 7 + 6*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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013610 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 A013610 → θ+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.