Hadamard-factor parent of A155495 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 8 + 24\,x + 9\,x^{2}\)

\(P_{2}(x) = 32 + 144\,x + 162\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	8	24	9															41
2	32	144	162	27														365
3	128	768	1440	864	81													3281
4	512	3840	10080	10800	4050	243												29525
5	2048	18432	62208	96768	68040	17496	729											265721
6	8192	86016	354816	725760	762048	381024	71442	2187										2391485
7	32768	393216	1916928	4866048	6842880	5225472	1959552	279936	6561									21523361
8	131072	1769472	9953280	30191616	53374464	55427328	32332608	9447840	1062882	19683								193710245
9	524288	7864320	50135040	176947200	377395200	498161664	400308480	184757760	43302600	3936600	59049							1743392201
10	2097152	34603008	246546432	992673792	2481684480	3985293312	4109833728	2642035968	990763488	190531440	14289858	177147						15690529805
11	8388608	150994944	1189085184	5379194880	15431565312	29238755328	36954537984	30743691264	16212493440	5043886848	810624672	51018336	531441					141214768241
12	33554432	654311424	5643436032	28339863552	91782512640	200610349056	300915523584	308834353152	212323617792	93672184320	24588948384	3353038416	179627058	1594323				1270932914165
13	134217728	2818572288	26424115200	145861115904	526467465216	1304723718144	2268440100864	2777681756160	2369709748224	1371937222656	514476458496	115550862336	13541116680	624974616	4782969			11438396227481
14	536870912	12079595520	122305904640	736100352000	2930245632000	8122640891904	16076060098560	22915874488320	23436689817600	16926498201600	8378616609792	2706012541440	526169105280	53569252800	2152336050	14348907		102945566047325
15	2147483648	51539607552	560493232128	3652869685248	15899767603200	48759287316480	108301878558720	176377345081344	210826045292544	183326995906560	113746067914752	48748314820608	13710463543296	2331333881856	208154810880	7346640384	43046721	926510094425921

Technical details

Node id: n-c51d3e880faa56cc · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-38d9a649e263b473

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       8,       24,         9;
   2:      32,      144,       162,        27;
   3:     128,      768,      1440,       864,         81;
   4:     512,     3840,     10080,     10800,       4050,        243;
   5:    2048,    18432,     62208,     96768,      68040,      17496,        729;
   6:    8192,    86016,    354816,    725760,     762048,     381024,      71442,       2187;
   7:   32768,   393216,   1916928,   4866048,    6842880,    5225472,    1959552,     279936,      6561;
   8:  131072,  1769472,   9953280,  30191616,   53374464,   55427328,   32332608,    9447840,   1062882,     19683;
   9:  524288,  7864320,  50135040, 176947200,  377395200,  498161664,  400308480,  184757760,  43302600,   3936600,    59049;
  10: 2097152, 34603008, 246546432, 992673792, 2481684480, 3985293312, 4109833728, 2642035968, 990763488, 190531440, 14289858, 177147;

%F Formula

The row polynomials satisfy P(n,x) = -9*x^2*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 8 + 24*x + 9*x^2, P(2,x) = 32 + 144*x + 162*x^2 + 27*x^3.

%t Mathematica

P[1, x_] := 8 + 24*x + 9*x^2;
P[2, x_] := 32 + 144*x + 162*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[-9*x^2*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 8 + 24*x + 9*x^2; v[2] = 32 + 144*x + 162*x^2 + 27*x^3; for(n = 3, N, my(i = n); v[i] = -9*x^2*v[i - 2] + (4 + 6*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023