A153648 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-30 - 60\,x + 15\,n + 30\,n x\right)\,P_{n-2}(x) + \left(-4 - 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 10 + 10\,x\)

\(P_{2}(x) = 50 + 100\,x + 8\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	10	10														20
2	50	100	8													158
3	250	750	516	16												1532
4	1250	5000	5652	1304	32											13238
5	6250	31250	51544	29632	3152	64										121892
6	31250	187500	400368	351472	105168	7392	128									1083278
7	156250	1093750	2886396	3494480	1849520	320640	16960	256								9818252
8	781250	6250000	19590812	30044872	22809680	7606240	898880	38272	512							88020518
9	3906250	35156250	127790064	238280448	238286304	121127040	26883840	2386944	85248	1024						793903412
10	19531250	195312500	807254808	1770538464	2196624864	1517210688	537692160	86082048	6100224	187904	2048					7136536958
11	97656250	1074218750	4976673076	12563495184	18657034656	16393165824	8146874496	2089867776	257251584	15149056	410624	4096				64271801372
12	488281250	5859375000	30069980772	85856057720	148506586560	158757307200	102661783680	37895480064	7368203520	730626560	36787200	890880	8192			578231368598
13	2441406250	31738281250	178735370184	569604697024	1125485663920	1420136314560	1136836273920	556868779008	157082174208	24142443520	1995458560	87736320	1921024	16384		5205156536132
14	12207031250	170898437500	1047778264448	3686620423376	8193461251888	11927801161760	11416591148160	7035695817216	2675289361152	594454739456	74746408960	5283368960	206163968	4120576	32768	46841037731438

Technical details

Node id: n-51e00fb280b8d256 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-835f2c52ab764e58

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:       2;
  1:      10,       10;
  2:      50,      100,         8;
  3:     250,      750,       516,        16;
  4:    1250,     5000,      5652,      1304,        32;
  5:    6250,    31250,     51544,     29632,      3152,        64;
  6:   31250,   187500,    400368,    351472,    105168,      7392,      128;
  7:  156250,  1093750,   2886396,   3494480,   1849520,    320640,    16960,     256;
  8:  781250,  6250000,  19590812,  30044872,  22809680,   7606240,   898880,   38272,   512;
  9: 3906250, 35156250, 127790064, 238280448, 238286304, 121127040, 26883840, 2386944, 85248, 1024;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (15*n-30+(30*n-60)*x)*P(n-2,x) + (2*n-4+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 10 + 10*x, P(2,x) = 50 + 100*x + 8*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 10*x;
P[2, x_] := 50 + 100*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n - 30 + (30*n - 60)*x)*P[n - 2, x] + (2*n - 4 + (2*n - 4)*x)*P[n - 1, 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] = 2; v[2] = 10 + 10*x; v[3] = 50 + 100*x + 8*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((15*n - 30 + (30*n - 60)*x)*v[i - 2] + (2*n - 4 + (2*n - 4)*x)*v[i - 1])/(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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check

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