A143491 → basis A013609 → D

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Submission text ?

Triangle begins:
  0:          14,           8;
  1:          94,          96,          24;
  2:         684,         952,         432,          64;
  3:        5508,        9400,        5880,        1600,         160;
  4:       49104,       97232,       75240,       28480,        5280,        384;
  5:      482256,     1071504,      968856,      456960,      118720,      16128,        896;
  6:     5185440,    12644064,    12903072,     7153216,     2329600,     446208,      46592,      2048;
  7:    60668640,   159826464,   179865984,   113093568,    43508640,   10499328,    1553664,    129024,     4608;
  8:   767940480,  2160592128,  2637871200,  1836765440,   804837600,  230412672,   43142400,   5099520,   345600,  10240;
  9: 10462227840, 31159459584, 40774101984, 30912273920, 15054802400, 4926974976, 1098480768, 164904960, 15966720, 901120, 22528;

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(4*n+12)*x+4*x^2)*P(n-2,x) + (2*n+7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 14 + 8*x, P(1,x) = 94 + 96*x + 24*x^2.

P[0, x_] := 14 + 8*x;
P[1, x_] := 94 + 96*x + 24*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (4*n + 12)*x + 4*x^2)*P[n - 2, x] + (2*n + 7 + 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] = 14 + 8*x; v[2] = 94 + 96*x + 24*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (4*n + 12)*x + 4*x^2)*v[i - 2] + (2*n + 7 + 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 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_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 ?

• A136124 → basis A013609 → 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.