A204420 → basis A013609 → basis A013609

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Submission text ?

Triangle begins:
  0:                 1;
  1:                 3,                  4;
  2:                45,                 96,                 48;
  3:              1575,               4260,               3600,                960;
  4:             99225,             312480,             346080,             161280,              26880;
  5:           9823275,           34507620,           45511200,           28425600,            8467200,            967680;
  6:        1404728325,         5366813760,         8026436880,         6067353600,         2461536000,         510935040,          42577920;
  7:      273922023375,      1119574556100,      1844229507120,      1600508669760,       795501907200,      227632204800,       34871316480,       2214051840;
  8:    69850115960625,    301926833208000,    537452997681600,    518783481216000,    298883506521600,   105776326656000,    22550117990400,    2656862208000,    132843110400;
  9: 22561587455281875, 102272175011506500, 194128342909300800, 204113868275116800, 131816649329164800, 54489831953356800, 14476478323507200, 2391574516531200, 223574954803200, 9033331507200;

The row polynomials satisfy P(n,x) = (4*n^2-1+(8*n-4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n^2 - 1 + (8*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; for(n = 1, N, my(i = n + 1); v[i] = (4*n^2 - 1 + (8*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of e…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
        • 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
        • Parent-row provers of polytool polytool ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A290319 → basis A013611 → basis A013609
• A028338 → x ↦ x+1 → basis A013611
• A161198 → basis A013609 → reverse
• A161198 → basis A013609

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.