A204420 → basis A013609 → basis A008277

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Submission text ?

Triangle begins:
  0:                 1;
  1:                 3,                  2;
  2:                45,                 60,                 12;
  3:              1575,               3150,               1260,                120;
  4:             99225,             264600,             158760,              30240,              1680;
  5:           9823275,           32744250,           26195400,            7484400,            831600,            30240;
  6:        1404728325,         5618913300,         5618913300,         2140538400,         356756400,         25945920,          665280;
  7:      273922023375,      1278302775750,      1533963330900,       730458729000,      162324162000,      17708090400,       908107200,       17297280;
  8:    69850115960625,    372533951790000,    521547532506000,    298027161432000,    82785322620000,   12041501472000,    926269344000,    35286451200,     518918400;
  9: 22561587455281875, 135369524731691250, 216591239570706000, 144394159713804000, 48131386571268000, 8751161194776000, 897554994336000, 51288856819200, 1508495788800, 17643225600;

The row polynomials satisfy P(n,x) = (4*n^2-1+(4*n-2)*x)*P(n-1,x) + (4*n-2)*x*d/dx 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 + (4*n - 2)*x)*P[n - 1, x] + (4*n - 2)*x*D[P[n - 1, x], 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 + (4*n - 2)*x)*v[i - 1] + (4*n - 2)*x*deriv(v[i - 1], x)); 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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 A008277
• A161198 → basis A008277
• A161198 → basis A008277 → reverse

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.