A008306 → D+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(n x^{2} - n^{3} x^{2}\right)\,P_{n-4}(x) + \left(2\,n x - 2\,n^{3} x\right)\,P_{n-3}(x) + \left(n + 2\,n x + 2\,n^{2} x - n^{3}\right)\,P_{n-2}(x) + \left(-1 + n + 2\,n^{2}\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 9 + 3\,x\)

\(P_{3}(x) = 44 + 20\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	9	3							12
3	44	20							64
4	250	160	15						425
5	1644	1344	210						3198
6	12348	12068	2695	105					27216
7	104544	117088	33992	2520					258144
8	986256	1230696	435960	47880	945				2701737
9	10265760	13994640	5794800	843920	34650				30933770
10	116915040	171646992	80620716	14563780	918225	10395			384675148
11	1446526080	2262596544	1179530352	252766800	21561540	540540			5163521856
12	19323757440	31934627712	18180522360	4478321848	480890410	19099080	135135		74417353985
13	277238626560	480881495808	295269169104	81692570960	10535224700	576816240	9459450		1146203362822
14	4251984710400	7699529848320	5049341135280	1541904283920	231044934120	16139923800	430404975	2027025	18790377267840

Technical details

Node id: n-61fe0d9f1ff3ff7f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9342e2e07a2452d6

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:        2;
  2:        9,        3;
  3:       44,       20;
  4:      250,      160,      15;
  5:     1644,     1344,     210;
  6:    12348,    12068,    2695,    105;
  7:   104544,   117088,   33992,   2520;
  8:   986256,  1230696,  435960,  47880,   945;
  9: 10265760, 13994640, 5794800, 843920, 34650;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n^3-n)*x^2*P(n-4,x) - (2*n^3-2*n)*x*P(n-3,x) - (n^3-n-(2*n^2+2*n)*x)*P(n-2,x) + (2*n^2+n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 9 + 3*x, P(3,x) = 44 + 20*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 9 + 3*x;
P[3, x_] := 44 + 20*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n)*x^2*P[n - 4, x] - (2*n^3 - 2*n)*x*P[n - 3, x] - (n^3 - n - (2*n^2 + 2*n)*x)*P[n - 2, x] + (2*n^2 + n - 1)*P[n - 1, 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] = 2; v[3] = 9 + 3*x; v[4] = 44 + 20*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - n)*x^2*v[i - 4] - (2*n^3 - 2*n)*x*v[i - 3] - (n^3 - n - (2*n^2 + 2*n)*x)*v[i - 2] + (2*n^2 + n - 1)*v[i - 1])/(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 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows; related to number of preorders.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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.