A060187 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 8\,x + 2\,n\right)\,P_{n}(x) = \left(15 - 234\,x + 1696\,x^{2} - 2656\,x^{3} + 768\,x^{4} - 6\,n + 124\,n x - 1288\,n x^{2} + 1440\,n x^{3} - 256\,n x^{4} - 16\,n^{2} x + 352\,n^{2} x^{2} - 192\,n^{2} x^{3} - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(10\,x - 196\,x^{2} + 928\,x^{3} - 1344\,x^{4} + 512\,x^{5} - 4\,n x + 112\,n x^{2} - 480\,n x^{3} + 384\,n x^{4} - 16\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-26 + 270\,x - 344\,x^{2} + 64\,x^{3} + 8\,n - 124\,n x + 80\,n x^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-14\,x + 72\,x^{2} - 64\,x^{3} + 4\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1								1
3	5	4							9
4	21	20							41
5	73	232	80						385
6	233	2184	976						3393
7	717	16044	19376	3904					40041
8	2173	101724	315696	88640					508233
9	6545	591184	3945696	2636032	354560				7534017
10	19665	3256976	41223648	64935168	12933376				122368833
11	59029	17346772	382998176	1224066176	533487872	51733504			2209691529
12	177125	90393636	3287947168	18960704128	18308096256	2767631360			43414949673
13	531417	464366904	26720883632	256223180032	482264473344	150506371072	11070525440		927250331841
14	1594297	2363285016	208867710000	3143417585920	10340994666240	6813984970752	816582578176		21326212390401
15	4782941	11953534588	1587313727184	35948453743040	191006973525760	237967012893696	56453759725568	3266330312704	526241802245481

Technical details

Node id: n-5dadc9a0b740e8b3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-81cf55bca2009298

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     1;
   3:     5,       4;
   4:    21,      20;
   5:    73,     232,       80;
   6:   233,    2184,      976;
   7:   717,   16044,    19376,     3904;
   8:  2173,  101724,   315696,    88640;
   9:  6545,  591184,  3945696,  2636032,   354560;
  10: 19665, 3256976, 41223648, 64935168, 12933376;

%F Formula

The row polynomials satisfy (2*n-7+8*x)*P(n,x) = -(6*n-15+(16*n^2-124*n+234)*x+(32*n^3-352*n^2+1288*n-1696)*x^2+(192*n^2-1440*n+2656)*x^3+(256*n-768)*x^4)*P(n-2,x) - ((4*n-10)*x+(16*n^2-112*n+196)*x^2-(64*n^2-480*n+928)*x^3-(384*n-1344)*x^4-512*x^5)*d/dx P(n-2,x) + (8*n-26+(16*n^2-124*n+270)*x+(80*n-344)*x^2+64*x^3)*P(n-1,x) + ((4*n-14)*x-(16*n-72)*x^2-64*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 15 + (16*n^2 - 124*n + 234)*x + (32*n^3 - 352*n^2 + 1288*n - 1696)*x^2 + (192*n^2 - 1440*n + 2656)*x^3 + (256*n - 768)*x^4)*P[n - 2, x] - ((4*n - 10)*x + (16*n^2 - 112*n + 196)*x^2 - (64*n^2 - 480*n + 928)*x^3 - (384*n - 1344)*x^4 - 512*x^5)*D[P[n - 2, x], x] + (8*n - 26 + (16*n^2 - 124*n + 270)*x + (80*n - 344)*x^2 + 64*x^3)*P[n - 1, x] + ((4*n - 14)*x - (16*n - 72)*x^2 - 64*x^3)*D[P[n - 1, x], x])/(2*n - 7 + 8*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(6*n - 15 + (16*n^2 - 124*n + 234)*x + (32*n^3 - 352*n^2 + 1288*n - 1696)*x^2 + (192*n^2 - 1440*n + 2656)*x^3 + (256*n - 768)*x^4)*v[i - 2] - ((4*n - 10)*x + (16*n^2 - 112*n + 196)*x^2 - (64*n^2 - 480*n + 928)*x^3 - (384*n - 1344)*x^4 - 512*x^5)*deriv(v[i - 2], x) + (8*n - 26 + (16*n^2 - 124*n + 270)*x + (80*n - 344)*x^2 + 64*x^3)*v[i - 1] + ((4*n - 14)*x - (16*n - 72)*x^2 - 64*x^3)*deriv(v[i - 1], x))/(2*n - 7 + 8*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 gamma_polynomial of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) =…?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

type_b_eulerian → γ → D+1

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.