A257611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-15 + 18\,x - 3\,x^{2} - 30\,n + 24\,n x + 2\,n x^{2} - 24\,n^{2} x + 12\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 8\,x^{2} - 2\,x^{3} - 12\,n x + 12\,n x^{2} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 2\,x + 16\,n - 8\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3														3
2	30	18													48
3	213	426	81												720
4	1308	5964	3924	324											11520
5	7431	65292	97938	29724	1215										201600
6	40314	620526	1762884	1241052	201570	4374									3870720
7	212505	5391846	26104635	34806180	13479615	1275030	15309								81285120
8	1099704	44118072	340082712	765735960	566804520	132354216	7697928	52488							1857945600
9	5623179	346216440	4053401028	14317862568	17897328210	8106802056	1211757540	44985432	177147						45984153600
10	28529238	2637195882	45309128472	238564508808	465330533460	357846763212	105721299768	10548783528	256763142	590490					1226244096000
11	144004317	19658599170	482942238885	3651216508920	10522381475130	12626857770156	6389628890610	1287845970360	88463696265	1440043170	1948617				35070581145600
12	724450260	144234392772	4965726023820	52397918316540	214379309991240	378805733104680	300131033987736	104795836633080	14897178071460	721171963860	7968952860	6377292			1071246842265600
13	3636600207	1045863262404	49667620765014	715417430148900	4031381589903945	10055423920391496	11731327907123412	6450210543846312	1609689217835025	165558735883380	5752247943222	43639202484	20726199		34815522373632000
14	18229236402	7517419248006	486228459225276	9392732101764996	71187668210159910	242746859055682386	398865148842196008	323662478740909848	128137802778287838	23481830254412490	1782837683826012	45104515488036	236980073226	66961566	1199796463337472000

Technical details

Node id: n-ea6477fe3049a666 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a1d3ab53fbcf8658

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:       3;
  2:      30,        18;
  3:     213,       426,         81;
  4:    1308,      5964,       3924,         324;
  5:    7431,     65292,      97938,       29724,        1215;
  6:   40314,    620526,    1762884,     1241052,      201570,       4374;
  7:  212505,   5391846,   26104635,    34806180,    13479615,    1275030,      15309;
  8: 1099704,  44118072,  340082712,   765735960,   566804520,  132354216,    7697928,    52488;
  9: 5623179, 346216440, 4053401028, 14317862568, 17897328210, 8106802056, 1211757540, 44985432, 177147;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = -(30*n+15+(24*n^2-24*n-18)*x+(8*n^3-12*n^2-2*n+3)*x^2)*P(n-2,x) - ((12*n+6)*x+(8*n^2-12*n-8)*x^2-(8*n^2-2)*x^3)*d/dx P(n-2,x) + (16*n-2+(8*n^2-8*n+2)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n + 15 + (24*n^2 - 24*n - 18)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*P[n - 2, x] - ((12*n + 6)*x + (8*n^2 - 12*n - 8)*x^2 - (8*n^2 - 2)*x^3)*D[P[n - 2, x], x] + (16*n - 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n + 15 + (24*n^2 - 24*n - 18)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*v[i - 2] - ((12*n + 6)*x + (8*n^2 - 12*n - 8)*x^2 - (8*n^2 - 2)*x^3)*deriv(v[i - 2], x) + (16*n - 2 + (8*n^2 - 8*n + 2)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.?
  - 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) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.?
    
    ↑ Liu–Wang theorem (above)

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.