type_b_eulerian → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(3 - 10\,x + 15\,x^{2} - 6\,n + 24\,n x - 46\,n x^{2} - 8\,n^{2} x + 36\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 8\,x^{2} + 6\,x^{3} - 4\,n x + 20\,n x^{2} - 16\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-10 + 18\,x + 8\,n - 24\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	6	2													8
3	23	46	3												72
4	76	460	228	4											768
5	237	3364	5046	948	5										9600
6	722	21086	70644	42172	3610	6									138240
7	2179	121314	779169	1038892	303285	13074	7								2257920
8	6552	663224	7455864	18700056	12426440	1989672	45864	8							41287680
9	19673	3512680	65123916	277653176	347066470	130247832	12294380	157384	9						836075520
10	59038	18232282	534902712	3627693128	7635462340	5441539692	1248106328	72929128	531342	10					18579456000
11	177135	93404854	4209242175	43347292392	142940099070	171528118884	75857761686	11224645800	420321843	1771350	11				449622835200
12	531428	474463940	32126733564	485535120828	2385102822120	4459731090984	3339143950968	971070241656	96380200692	2372319700	5845708	12			11771943321600
13	1594309	2396765388	239832749058	5183693337740	36553752360915	100908354980952	117726414144444	58486003777464	11663310009915	799442496860	13182209634	19131708	13		331576403558400
14	4782954	12063542390	1761517649292	53368557013284	525203992741390	2057505162915042	3531792424333320	2743340217220056	945367186934502	133421392533210	6458898047404	72381254340	62178402	14	9998303861145600

Technical details

Node id: n-3e0d99f9f1c80693 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a7d11e539b3a8882

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:     0;
  1:     1;
  2:     6,       2;
  3:    23,      46,        3;
  4:    76,     460,      228,         4;
  5:   237,    3364,     5046,       948,         5;
  6:   722,   21086,    70644,     42172,      3610,         6;
  7:  2179,  121314,   779169,   1038892,    303285,     13074,        7;
  8:  6552,  663224,  7455864,  18700056,  12426440,   1989672,    45864,      8;
  9: 19673, 3512680, 65123916, 277653176, 347066470, 130247832, 12294380, 157384, 9;

%F Formula

The row polynomials satisfy (2*n-3)*P(n,x) = -(6*n-3+(8*n^2-24*n+10)*x+(8*n^3-36*n^2+46*n-15)*x^2)*P(n-2,x) - ((4*n-2)*x+(8*n^2-20*n+8)*x^2-(8*n^2-16*n+6)*x^3)*d/dx P(n-2,x) + (8*n-10+(8*n^2-24*n+18)*x)*P(n-1,x) + ((4*n-6)*x-(4*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 3 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*P[n - 2, x] - ((4*n - 2)*x + (8*n^2 - 20*n + 8)*x^2 - (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], x] + (8*n - 10 + (8*n^2 - 24*n + 18)*x)*P[n - 1, x] + ((4*n - 6)*x - (4*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 3 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*v[i - 2] - ((4*n - 2)*x + (8*n^2 - 20*n + 8)*x^2 - (8*n^2 - 16*n + 6)*x^3)*deriv(v[i - 2], x) + (8*n - 10 + (8*n^2 - 24*n + 18)*x)*v[i - 1] + ((4*n - 6)*x - (4*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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 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 ?

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.