A204420 → basis A007318 → reverse

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Submission text ?

Triangle begins:
  0:        1;
  1:        1,          1;
  2:        3,         12,           9;
  3:       15,        135,         345,           225;
  4:      105,       1680,        9030,         18480,          11025;
  5:      945,      23625,      217350,        897750,        1596105,          893025;
  6:    10395,     374220,     5249475,      36174600,      126184905,       202951980,       108056025;
  7:   135135,    6621615,   131486355,    1357431075,     7753911165,     23963624685,     35703612945,      18261468225;
  8:  2027025,  129729600,  3462158700,   49945896000,   421730659350,   2104084382400,   5927369748300,    8307234936000,    4108830350625;
  9: 34459425, 2791213425, 96348552300, 1849644096300, 21603785152950, 157649595052950, 708845672234700, 1854232851170700, 2470641012464625, 1187451971330625;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + (4*n^2 - 4*n + 1)*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] = (2*n - 1 + (4*n^2 - 4*n + 1)*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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 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)
    • Affine, reversal and Möbius substitutions (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 ?

• A204420 → basis A007318
• A028338: Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
• A109692: Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).
• A290319 → basis A013609
• A290319 → basis A013609 → 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.