A308737 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-8 - 10\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(-10\,x - 4\,x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2}\right)\,P_{n-2}''(x) + \left(6 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	16	29	7													52
3	64	215	121	15												415
4	256	1445	1435	404	31											3571
5	1024	9263	14545	7007	1198	63										33100
6	4096	57869	135667	99820	28595	3303	127									329477
7	16384	356135	1204441	1273615	544670	103812	8671	255								3507983
8	65536	2171765	10361995	15168804	9098691	2547258	347004	21974	511							39783538
9	262144	13168223	87298465	172545087	139638198	53906265	10656282	1090562	54220	1023						478620469
10	1048576	79553309	724966627	1900772060	2021579395	1034411301	278437929	40961580	3267066	131009	2047					6085130899
11	4194304	479476055	5959829161	20461517615	28066206190	18528878214	6525275757	1294479285	147298800	9418565	311221	4095				81476889262
12	16777216	2885422085	48641602555	216555142804	377755629251	315570145176	141477700078	36355279818	5535572757	501864935	26309569	729000	8191			1145322183435
13	67108864	17346618383	394919910385	2262940191967	4966426863398	5173765475307	2894941181704	937402878984	183432497820	22113460226	1635256051	71577417	1687450	16383		16855064724339
14	268435456	104215522349	3194118034387	23419785780300	64124602464195	82367724698259	56668123563051	22663763854160	5543612986912	853385751218	83475510118	5131677096	190409141	3866507	32767	259028402585916

Technical details

Node id: n-8b21094149bba3a5 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-b534cd5aa4997db4

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:      4,        3;
  2:     16,       29,        7;
  3:     64,      215,      121,        15;
  4:    256,     1445,     1435,       404,        31;
  5:   1024,     9263,    14545,      7007,      1198,       63;
  6:   4096,    57869,   135667,     99820,     28595,     3303,      127;
  7:  16384,   356135,  1204441,   1273615,    544670,   103812,     8671,     255;
  8:  65536,  2171765, 10361995,  15168804,   9098691,  2547258,   347004,   21974,   511;
  9: 262144, 13168223, 87298465, 172545087, 139638198, 53906265, 10656282, 1090562, 54220, 1023;

%F Formula

The row polynomials satisfy P(n,x) = -(8+10*x+2*x^2)*P(n-2,x) - (10*x+4*x^2)*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 10*x + 2*x^2)*P[n - 2, x] - (10*x + 4*x^2)*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 10*x + 2*x^2)*v[i - 2] - (10*x + 4*x^2)*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (6 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); 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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741