A257615 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,n\right)\,P_{n}(x) = \left(-105 + 30\,x + 3\,x^{2} - 70\,n - 40\,n x + 2\,n x^{2} - 40\,n^{2} x - 12\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-30\,x + 24\,x^{2} + 6\,x^{3} - 20\,n x + 4\,n x^{2} + 16\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(22 + 2\,x + 24\,n + 8\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 5\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	5														5
2	70	50													120
3	715	1430	375												2520
4	6380	25740	19140	2500											53760
5	52785	372020	558030	211140	15625										1209600
6	416370	4720590	12276660	9441180	2081850	93750									29030400
7	3180215	54976410	227094765	302793020	137441025	19081290	546875								745113600
8	23745880	602915000	3735011640	7872618520	6225019400	1808745000	166221160	3125000							20437401600
9	174424285	6328578440	56459460540	177044273560	221305341950	112918921080	22150024540	1395394280	17578125						597793996800
10	1265891870	64283025930	801418551480	3581323328520	6639160258500	5371984992780	1869976620120	257132103720	11393026830	97656250					18598035456000
11	9105383715	636778259390	10840519383075	66859806574920	175690524608550	210828629530260	117004661506110	28908051688200	2865502167255	91053837150	537109375				613735170048000
12	65056045380	6186273520260	141214563162780	1172712028200060	4223278275282600	7168173404028840	5912589585395640	2345424056400120	423643689488340	30931367601300	715616499180	2929687500			21424936845312000
13	462472395785	59189488132860	1785345451800330	19575836303592700	94132864869490575	218149692544933560	254507974635755820	150612583791184920	44045631683083575	5951151506001100	325542184730730	5549668749420	15869140625		789151840468992000
14	3275148567370	559528792151270	22035974239184460	313997387006716100	1974798266770648750	6080692967975033010	9671303036158721160	8107590623966710680	3554636880187167750	784993467516790250	80798572210343020	3357172752907620	42576931375810	85449218750	30594809815105536000

Technical details

Node id: n-2e0e4d5e661bfdfc · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-de1cf4a5947ef456

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:         5;
  2:        70,         50;
  3:       715,       1430,         375;
  4:      6380,      25740,       19140,         2500;
  5:     52785,     372020,      558030,       211140,        15625;
  6:    416370,    4720590,    12276660,      9441180,      2081850,        93750;
  7:   3180215,   54976410,   227094765,    302793020,    137441025,     19081290,      546875;
  8:  23745880,  602915000,  3735011640,   7872618520,   6225019400,   1808745000,   166221160,    3125000;
  9: 174424285, 6328578440, 56459460540, 177044273560, 221305341950, 112918921080, 22150024540, 1395394280, 17578125;

%F Formula

The row polynomials satisfy (2*n+1)*P(n,x) = -(70*n+105+(40*n^2+40*n-30)*x+(8*n^3+12*n^2-2*n-3)*x^2)*P(n-2,x) - ((20*n+30)*x+(8*n^2-4*n-24)*x^2-(8*n^2+16*n+6)*x^3)*d/dx P(n-2,x) + (24*n+22+(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) = 5.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[(-(70*n + 105 + (40*n^2 + 40*n - 30)*x + (8*n^3 + 12*n^2 - 2*n - 3)*x^2)*P[n - 2, x] - ((20*n + 30)*x + (8*n^2 - 4*n - 24)*x^2 - (8*n^2 + 16*n + 6)*x^3)*D[P[n - 2, x], x] + (24*n + 22 + (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] = 5; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n + 105 + (40*n^2 + 40*n - 30)*x + (8*n^3 + 12*n^2 - 2*n - 3)*x^2)*v[i - 2] - ((20*n + 30)*x + (8*n^2 - 4*n - 24)*x^2 - (8*n^2 + 16*n + 6)*x^3)*deriv(v[i - 2], x) + (24*n + 22 + (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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.?
  - 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) = 2*x + 5.?
    
    ↑ 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.