A123519 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

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

\(P_{0}(x) = 11 + 18\,x + 4\,x^{2}\)

\(P_{1}(x) = 41 + 128\,x + 68\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	11	18	4								33
1	41	128	68	8							245
2	153	832	796	216	16						2013
3	571	5170	7964	3744	624	32					18105
4	2131	31330	73256	53152	14960	1696	64				176589
5	7953	187072	640744	674288	285200	53632	4416	128			1853433
6	29681	1107008	5426840	7971216	4756016	1319552	177856	11136	256		20799561
7	110771	6514338	44987992	89891552	72729424	27932128	5480384	556032	27392	512	248230525

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7d6ae414bdaf021b · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-d62365dbb465cfbd

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 2)
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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      11,        18,          4;
  1:      41,       128,         68,           8;
  2:     153,       832,        796,         216,          16;
  3:     571,      5170,       7964,        3744,         624,         32;
  4:    2131,     31330,      73256,       53152,       14960,       1696,         64;
  5:    7953,    187072,     640744,      674288,      285200,      53632,       4416,       128;
  6:   29681,   1107008,    5426840,     7971216,     4756016,    1319552,     177856,     11136,      256;
  7:  110771,   6514338,   44987992,    89891552,    72729424,   27932128,    5480384,    556032,    27392,     512;
  8:  413403,  38200562,  367505772,   980920288,  1047780176,  535189088,  143372544,  20958208,  1659648,   66048,   1024;
  9: 1542841, 223515840, 2971459308, 10454322872, 14462473264, 9560275456, 3358858496, 663436800, 75081984, 4771840, 156672, 2048;

%F Formula

The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 11 + 18*x + 4*x^2, P(1,x) = 41 + 128*x + 68*x^2 + 8*x^3.

%t Mathematica

P[0, x_] := 11 + 18*x + 4*x^2;
P[1, x_] := 41 + 128*x + 68*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 2*x)*P[n - 1, x] + 2*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] = 11 + 18*x + 4*x^2; v[2] = 41 + 128*x + 68*x^2 + 8*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 2*x)*v[i - 1] + 2*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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-ad85cdc8ea03f732
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-ad85cdc8ea03f732
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-ad85cdc8ea03f732

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A008277 e-ad85cdc8ea03f732

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8d057b494ddd47e8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-2a4594def0262eef root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023