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	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

All 15 rows

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