A085841 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-15 + 30\,x - 8\,x^{2} + 50\,n - 76\,n x + 16\,n x^{2} - 48\,n^{2} + 32\,n^{2} x + 16\,n^{3}\right)\,P_{n}(x) = \left(3 - 114\,x + 448\,x^{2} - 160\,x^{3} + 2\,n + 316\,n x - 832\,n x^{2} + 192\,n x^{3} - 480\,n^{2} x + 384\,n^{2} x^{2} + 16\,n^{3} + 192\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x - 104\,x^{2} + 256\,x^{3} - 128\,x^{4} - 32\,n x + 192\,n x^{2} - 256\,n x^{3} + 32\,n^{2} x - 128\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	4														11
2	45	72	16													133
3	147	812	528	64												1551
4	345	4368	8928	3328	256											17225
5	671	15444	70752	77440	19200	1024										184531
6	1157	42328	350064	841984	582400	104448	4096									1926477
7	1835	97916	1283568	5536960	8270080	3978240	544768	16384								19729751
8	2737	200736	3833024	26138112	69588480	71163904	25346048	2752512	65536							199091089
9	3895	375972	9860544	97819904	410628608	744192000	555663360	153157632	13565952	262144						1985530011
10	5341	656488	22651344	308426240	1881152000	5362151424	7056924672	4026925056	887685120	65536000	1048576					19613162261
11	7107	1083852	47608528	852492608	7131840000	29501093888	60919226368	60958605312	27507228672	4974182400	311427072	4194304				192208990111
12	9225	1709360	93144480	2122771200	23344889600	132366950400	396187975680	620612157440	488713420800	179070566400	27105689600	1459617792	16777216			1871095679193
13	11727	2595060	171793440	4856142720	67953849600	506303585280	2075878932480	4707860152320	5790694440960	3685869158400	1120099368960	144275668992	6761218048	67108864		18110794026851
14	14645	3814776	301569840	10355255040	179701966080	1703251200000	9173029662720	28479138693120	50657774469120	50257149296640	26416801382400	6774086172672	752676241408	31004295168	268435456	174435542469085

Technical details

Node id: n-9255708251fbccb1 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-dceccde3bfe54822

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:    1;
  1:    7,      4;
  2:   45,     72,      16;
  3:  147,    812,     528,       64;
  4:  345,   4368,    8928,     3328,       256;
  5:  671,  15444,   70752,    77440,     19200,      1024;
  6: 1157,  42328,  350064,   841984,    582400,    104448,      4096;
  7: 1835,  97916, 1283568,  5536960,   8270080,   3978240,    544768,     16384;
  8: 2737, 200736, 3833024, 26138112,  69588480,  71163904,  25346048,   2752512,    65536;
  9: 3895, 375972, 9860544, 97819904, 410628608, 744192000, 555663360, 153157632, 13565952, 262144;

%F Formula

The row polynomials satisfy (16*n^3-48*n^2+50*n-15+(32*n^2-76*n+30)*x+(16*n-8)*x^2)*P(n,x) = (16*n^3+2*n+3+(192*n^3-480*n^2+316*n-114)*x+(384*n^2-832*n+448)*x^2+(192*n-160)*x^3)*P(n-1,x) + ((32*n^2-32*n+12)*x-(128*n^2-192*n+104)*x^2-(256*n-256)*x^3-128*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 2*n + 3 + (192*n^3 - 480*n^2 + 316*n - 114)*x + (384*n^2 - 832*n + 448)*x^2 + (192*n - 160)*x^3)*P[n - 1, x] + ((32*n^2 - 32*n + 12)*x - (128*n^2 - 192*n + 104)*x^2 - (256*n - 256)*x^3 - 128*x^4)*D[P[n - 1, x], x])/(16*n^3 - 48*n^2 + 50*n - 15 + (32*n^2 - 76*n + 30)*x + (16*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 2*n + 3 + (192*n^3 - 480*n^2 + 316*n - 114)*x + (384*n^2 - 832*n + 448)*x^2 + (192*n - 160)*x^3)*v[i - 1] + ((32*n^2 - 32*n + 12)*x - (128*n^2 - 192*n + 104)*x^2 - (256*n - 256)*x^3 - 128*x^4)*deriv(v[i - 1], x))/(16*n^3 - 48*n^2 + 50*n - 15 + (32*n^2 - 76*n + 30)*x + (16*n - 8)*x^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: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023