A142459 → D

nonnegative coefficients

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

Recurrence

\(\left(-11 + 4\,n\right)\,P_{n}(x) = \left(35 - 210\,x + 1155\,x^{2} - 20\,n + 176\,n x - 1388\,n x^{2} - 32\,n^{2} x + 528\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(28\,x - 336\,x^{2} + 308\,x^{3} - 16\,n x + 304\,n x^{2} - 288\,n x^{3} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-62 + 242\,x + 24\,n - 176\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-44\,x + 44\,x^{2} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	10	2													12
4	59	118	3												180
5	308	2124	924	4											3360
6	1557	27124	40686	6228	5										75600
7	7806	297054	1057836	594108	39030	6									1995840
8	39055	3001338	21326745	28435660	7503345	234330	7								60540480
9	195304	28964792	371789832	966812440	619649720	86894376	1367128	8							2075673600
10	976553	272010760	5919447516	26845926776	33557408470	11838895032	952037660	7812424	9						79394515200
11	4882802	2512549338	88785285768	653695672392	1409411155740	980543508588	207165666792	10050197352	43945218	10					3352212864000
12	24414051	22974271390	1278579907215	14545857480360	50025624032790	60030748839348	25455250590630	3409546419240	103384221255	244140510	11				154872234316800
13	122070300	208770394692	17896610855940	303537093083580	1577825438351640	3001537441967400	2208955613692296	607074186167160	53689832567820	1043851973460	1342773300	12			7771770303897600
14	610351549	1889919879228	245495320400778	6043030051313900	45655239295082115	129946761303822072	151604554854459084	73048382872131384	13596817615456275	818317734669260	10394559335754	7324218588	13		420970891461120000
15	3051757798	17069093364854	3319008757058004	116151921720912164	1238250165069992290	5056616508680803554	8793064181558626872	6742155344907738072	2228850297125986122	290379804302280410	12169698775879348	102414560189124	39672851374	14	24481076457277440000

Technical details

Node id: n-ad424251fd593cd0 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6bc0895ac2a350b9

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 -2)
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:      0;
   2:      1;
   3:     10,         2;
   4:     59,       118,          3;
   5:    308,      2124,        924,           4;
   6:   1557,     27124,      40686,        6228,           5;
   7:   7806,    297054,    1057836,      594108,       39030,           6;
   8:  39055,   3001338,   21326745,    28435660,     7503345,      234330,         7;
   9: 195304,  28964792,  371789832,   966812440,   619649720,    86894376,   1367128,       8;
  10: 976553, 272010760, 5919447516, 26845926776, 33557408470, 11838895032, 952037660, 7812424, 9;

%F Formula

The row polynomials satisfy (4*n-11)*P(n,x) = -(20*n-35+(32*n^2-176*n+210)*x+(64*n^3-528*n^2+1388*n-1155)*x^2)*P(n-2,x) - ((16*n-28)*x+(64*n^2-304*n+336)*x^2-(64*n^2-288*n+308)*x^3)*d/dx P(n-2,x) + (24*n-62+(32*n^2-176*n+242)*x)*P(n-1,x) + ((16*n-44)*x-(16*n-44)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n - 35 + (32*n^2 - 176*n + 210)*x + (64*n^3 - 528*n^2 + 1388*n - 1155)*x^2)*P[n - 2, x] - ((16*n - 28)*x + (64*n^2 - 304*n + 336)*x^2 - (64*n^2 - 288*n + 308)*x^3)*D[P[n - 2, x], x] + (24*n - 62 + (32*n^2 - 176*n + 242)*x)*P[n - 1, x] + ((16*n - 44)*x - (16*n - 44)*x^2)*D[P[n - 1, x], x])/(4*n - 11)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(20*n - 35 + (32*n^2 - 176*n + 210)*x + (64*n^3 - 528*n^2 + 1388*n - 1155)*x^2)*v[i - 2] - ((16*n - 28)*x + (64*n^2 - 304*n + 336)*x^2 - (64*n^2 - 288*n + 308)*x^3)*deriv(v[i - 2], x) + (24*n - 62 + (32*n^2 - 176*n + 242)*x)*v[i - 1] + ((16*n - 44)*x - (16*n - 44)*x^2)*deriv(v[i - 1], x))/(4*n - 11)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,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: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257618 → D

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.