type_b_eulerian → γ → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 - 16\,x + n - 8\,n x + 32\,n x^{2} + 8\,n^{2} x + 16\,n^{2} x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 8\,x^{2} + 2\,n x - 16\,n x^{2} + 32\,n x^{3} - 8\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 8\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	1	8			9
3	1	40			41
4	1	144	240		385
5	1	464	2928		3393
6	1	1432	22992	15616	40041
7	1	4344	149328	354560	508233

All 15 rows

Technical details

Node id: n-0c7814fec01dfe29 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-ade430673929d22a

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 1;
  2: 1,     8;
  3: 1,    40;
  4: 1,   144,     240;
  5: 1,   464,    2928;
  6: 1,  1432,   22992,    15616;
  7: 1,  4344,  149328,   354560;
  8: 1, 13088,  876960,  4871168,  1772800;
  9: 1, 39328, 4855968, 52806656, 64666880;

%F Formula

The row polynomials satisfy n*P(n,x) = (n+1+(8*n^2-8*n-16)*x-(16*n^3-16*n^2-32*n)*x^2)*P(n-2,x) + ((2*n+2)*x-(8*n^2+16*n+8)*x^2+(32*n^2+32*n)*x^3)*d/dx P(n-2,x) - (1-(8*n^2-8*n)*x)*P(n-1,x) + (2*n*x-8*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (8*n^2 - 8*n - 16)*x - (16*n^3 - 16*n^2 - 32*n)*x^2)*P[n - 2, x] + ((2*n + 2)*x - (8*n^2 + 16*n + 8)*x^2 + (32*n^2 + 32*n)*x^3)*D[P[n - 2, x], x] - (1 - (8*n^2 - 8*n)*x)*P[n - 1, x] + (2*n*x - 8*n*x^2)*D[P[n - 1, x], x])/(n)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n + 1 + (8*n^2 - 8*n - 16)*x - (16*n^3 - 16*n^2 - 32*n)*x^2)*v[i - 2] + ((2*n + 2)*x - (8*n^2 + 16*n + 8)*x^2 + (32*n^2 + 32*n)*x^3)*deriv(v[i - 2], x) - (1 - (8*n^2 - 8*n)*x)*v[i - 1] + (2*n*x - 8*n*x^2)*deriv(v[i - 1], x))/(n)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from gamma_polynomial of type_b_eulerian?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from type_b_eulerian?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from type_b_eulerian?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A060187 → γ → θ+1

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.