A257606 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-60 - 24\,x - 6\,x^{2} - 20\,n - 32\,n x - 11\,n x^{2} - 8\,n^{2} x - 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 6\,x^{2} + 6\,x^{3} - 4\,n x - n x^{2} + 5\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(22 + 8\,x + 9\,n + 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	4														4
2	40	32													72
3	296	592	192												1080
4	1928	7104	5784	1024											15840
5	11688	69616	104424	46752	5120										237600
6	67656	604704	1461936	1209408	338280	24576									3706560
7	379240	4846032	17490000	23320000	12115080	2275440	114688								60540480
8	2076424	36660992	187851432	373120000	313085720	109982976	14534968	524288							1037836800
9	11168552	265647280	1864874904	5239177184	6548971480	3729749808	929765480	89348416	2359296						18681062400
10	59250632	1861928928	17437304448	66778416192	117881486640	100167624288	40687043712	7447715712	533255688	10485760					352864512000
11	310933224	12712090000	155575851840	789974461440	1895663582160	2274796298592	1382455307520	414868938240	57204405000	3109332240	46137344				6986717337600
12	1617580680	84978670272	1336916717880	8809009320960	27923121084240	45495925971840	39092369517936	17618018641920	4010750153640	424893351360	17793387480	201326592			144815595724800
13	8356338856	558399442032	11142969100872	93645297677600	383443229572560	823544458030368	960801867702096	613509167316096	210701919774600	37143230336240	3071196931176	100276066272	872415232		3137671240704000
14	42922544968	3617799495584	90564771151824	957164471303744	4972725153414040	13757027086148544	21137641089446112	18342702781531392	8950905276145272	2392911178259360	332070827556688	21706796973504	557993084584	3758096384	70959641905152000

Technical details

Node id: n-2ba9a596f53694d1 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-2603e33c60dc5afd

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:        4;
  2:       40,        32;
  3:      296,       592,        192;
  4:     1928,      7104,       5784,       1024;
  5:    11688,     69616,     104424,      46752,       5120;
  6:    67656,    604704,    1461936,    1209408,     338280,      24576;
  7:   379240,   4846032,   17490000,   23320000,   12115080,    2275440,    114688;
  8:  2076424,  36660992,  187851432,  373120000,  313085720,  109982976,  14534968,   524288;
  9: 11168552, 265647280, 1864874904, 5239177184, 6548971480, 3729749808, 929765480, 89348416, 2359296;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -(20*n+60+(8*n^2+32*n+24)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((4*n+12)*x+(n^2+n-6)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (9*n+22+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n + 60 + (8*n^2 + 32*n + 24)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((4*n + 12)*x + (n^2 + n - 6)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (9*n + 22 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n + 60 + (8*n^2 + 32*n + 24)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((4*n + 12)*x + (n^2 + n - 6)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (9*n + 22 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 1], x))/(n + 2)); 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.?
    
    ↑ 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.