A153649 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-70 - 280\,x + 35\,n + 140\,n x\right)\,P_{n-2}(x) + \left(-4 - 8\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 14 + 28\,x\)

\(P_{2}(x) = 98 + 392\,x + 32\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	14	28														42
2	98	392	32													522
3	686	4116	5552	128												10482
4	4802	38416	83568	26944	512											154242
5	33614	336140	1091360	1169920	126720	2048										2759802
6	235298	2823576	11830400	19347840	8723200	582656	8192									43551162
7	1647086	23059204	120212304	279754880	263061760	53870592	2633728	32768								744272322
8	11529602	184473632	1142026064	3373789376	4659152640	2401629184	302608384	11747328	131072							12087087282
9	80707214	1452729852	10457665856	37936826368	71186307584	62153986048	17845796864	1603796992	51838976	524288						202770180042
10	564950498	11299009960	92523471840	395670593280	929520775680	1145391593472	631126917120	117367111680	8168079360	226754560	2097152					3331861354602
11	3954653486	87002376692	799643467760	3953297216640	11244400381440	18150338863104	15169981145088	5313786347520	712150548480	40401633280	984612864	8388608				55475949634962
12	27682574402	664381785648	6767479351088	37906925126720	126229099837440	251499893618688	285385582927872	163773114580992	39396724899840	4080872980480	195394797568	4248829952	33554432			915931434865122
13	193778020814	5038228541164	56360339953248	353199255726592	1351101523861760	3217394100197376	4648767521243136	3778868443348992	1502741265973248	266863779512320	22404971626496	928216252416	18236833792	134217728		15203879795309082
14	1356446145698	37980492079544	462748821349952	3206029359853696	13839988082681088	38313764549737472	67365591952367616	71838847591907328	42428077718175744	12203064663212032	1690426096484352	118980299718656	4345349275648	77913391104	536870912	251511279873250842

Technical details

Node id: n-276f453ad85ebcdf · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-681a8e8c39404ee6

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:       14,         28;
  2:       98,        392,          32;
  3:      686,       4116,        5552,         128;
  4:     4802,      38416,       83568,       26944,         512;
  5:    33614,     336140,     1091360,     1169920,      126720,        2048;
  6:   235298,    2823576,    11830400,    19347840,     8723200,      582656,        8192;
  7:  1647086,   23059204,   120212304,   279754880,   263061760,    53870592,     2633728,      32768;
  8: 11529602,  184473632,  1142026064,  3373789376,  4659152640,  2401629184,   302608384,   11747328,   131072;
  9: 80707214, 1452729852, 10457665856, 37936826368, 71186307584, 62153986048, 17845796864, 1603796992, 51838976, 524288;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (35*n-70+(140*n-280)*x)*P(n-2,x) + (2*n-4+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 14 + 28*x, P(2,x) = 98 + 392*x + 32*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 14 + 28*x;
P[2, x_] := 98 + 392*x + 32*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((35*n - 70 + (140*n - 280)*x)*P[n - 2, x] + (2*n - 4 + (4*n - 8)*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] = 14 + 28*x; v[3] = 98 + 392*x + 32*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((35*n - 70 + (140*n - 280)*x)*v[i - 2] + (2*n - 4 + (4*n - 8)*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+1)*prime(j)*T(n-2, k-1) with j=4, 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