A159854 → basis A008277

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1 + 3\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	3	1															5
1	2	10	7	1														20
2	4	32	38	12	1													87
3	8	100	184	98	18	1												409
4	16	308	836	674	206	25	1											2066
5	32	940	3652	4206	1910	381	33	1										11155
6	64	2852	15548	24682	15666	4577	645	42	1									64077
7	128	8620	65044	138958	118678	47705	9737	1023	52	1								389946
8	256	25988	268796	759834	851026	452613	125601	18944	1543	63	1							2504665
9	512	78220	1101172	4067966	5865990	4019317	1457421	296097	34374	2236	75	1						16923381
10	1024	235172	4482908	21441002	39263906	34001209	15678685	4122294	639837	58970	3136	88	1					119928232
11	2048	706540	18166804	111687918	257024438	277272369	159430689	52779331	10520664	1288507	96602	4280	102	1				888980293
12	4096	2121668	73373756	576606394	1653834546	2197931021	1552717881	634444668	157985971	24694241	2447731	152242	5708	117	1			6876320041
13	8192	6369100	295616692	2956405726	10499613670	17039351693	14619674069	7262719893	2214304378	429622622	54067013	4426877	232154	7463	133	1		55382419676
14	16384	19115492	1188835868	15077645322	65954087746	129775075521	133996744245	79984153106	29405763673	6940153220	1078426778	111616414	7677033	344099	9591	150	1	463539664643

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Technical details

Node id: n-9a6b0e2632270c41 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-c748c34fca57e6cb

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 ?

A011966: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:   1,     3,       1;
  1:   2,    10,       7,       1;
  2:   4,    32,      38,      12,       1;
  3:   8,   100,     184,      98,      18,       1;
  4:  16,   308,     836,     674,     206,      25,       1;
  5:  32,   940,    3652,    4206,    1910,     381,      33,      1;
  6:  64,  2852,   15548,   24682,   15666,    4577,     645,     42,     1;
  7: 128,  8620,   65044,  138958,  118678,   47705,    9737,   1023,    52,    1;
  8: 256, 25988,  268796,  759834,  851026,  452613,  125601,  18944,  1543,   63,  1;
  9: 512, 78220, 1101172, 4067966, 5865990, 4019317, 1457421, 296097, 34374, 2236, 75, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + 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 + 3*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] + 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-94a89e959b0863ff
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-94a89e959b0863ff
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-94a89e959b0863ff

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-94a89e959b0863ff

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Riordan array (1-x,x/(1-x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-b62074ba45057e82 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[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