A204420 → basis A007318 → x ↦ x+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:                1;
  1:                2,                1;
  2:               24,               18,                3;
  3:              720,              660,              180,               15;
  4:            40320,            42000,            14700,             2100,             105;
  5:          3628800,          4142880,          1701000,           321300,           28350,            945;
  6:        479001600,        586776960,        270103680,         61122600,         7276500,         436590,         10395;
  7:      87178291200,     113020427520,      56786970240,      14635661040,      2118916800,      174053880,       7567560,       135135;
  8:   20922789888000,   28432576972800,   15324179270400,    4364363203200,    728074947600,    73556683200,    4427022600,    145945800,    2027025;
  9: 6402373705728000, 9056055981772800, 5172552665280000, 1596006187776000, 296985108420000, 34885619168400, 2605132530000, 119918799000, 3101348250, 34459425;

The row polynomials satisfy P(n,x) = (4*n^2-2*n+(2*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[(4*n^2 - 2*n + (2*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] = (4*n^2 - 2*n + (2*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 (−∞, -1]
  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), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (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)
• 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) (above)
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A109692 → x ↦ x/(x+1)
• A204420: Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, …
• A028338 → x ↦ x+1
• A225471 → basis A013609
• A028338 → x ↦ x+1 → reverse
• A204420 → reverse
• A225471 → 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.