A204420 → basis A007318 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-27 + 18\,x - 3\,x^{2} + 108\,n - 60\,n x + 8\,n x^{2} - 144\,n^{2} + 56\,n^{2} x - 4\,n^{2} x^{2} + 80\,n^{3} - 16\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 2\,x - 12\,n + 4\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	21	18	3						42
3	570	615	180	15					1380
4	29505	36540	14070	2100	105				82320
5	2489130	3391605	1549800	311850	28350	945			7771680
6	311008005	455321790	234708705	57172500	7120575	436590	10395		1065778560
7	53965081170	83630862315	47225358180	13183635465	2014862850	171216045	7567560	135135	200198718720

All 15 rows

Technical details

Node id: n-2fc865e4fc7ead86 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-54feb342b7863b00

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:                1;
  1:                2,                1;
  2:               21,               18,                3;
  3:              570,              615,              180,               15;
  4:            29505,            36540,            14070,             2100,             105;
  5:          2489130,          3391605,          1549800,           311850,           28350,            945;
  6:        311008005,        455321790,        234708705,         57172500,         7120575,         436590,         10395;
  7:      53965081170,      83630862315,      47225358180,      13183635465,      2014862850,      171216045,       7567560,       135135;
  8:   12416065286625,   20161974432600,   12239622895500,    3791007019800,    671460139350,    70718848200,    4370265900,    145945800,    2027025;
  9: 3658092983795250, 6179106714806025, 3980769867874800, 1339444052446500, 265668520817700, 32701649730750, 2524083962400, 118678259700, 3101348250, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(16*n^3-56*n^2+60*n-18)*x+(4*n^2-8*n+3)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (16*n^3 - 56*n^2 + 60*n - 18)*x + (4*n^2 - 8*n + 3)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (4*n - 2)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (16*n^3 - 56*n^2 + 60*n - 18)*x + (4*n^2 - 8*n + 3)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (4*n - 2)*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)

polytool c48629374597

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_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 ?

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.