A257619 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 - 18\,x + 18\,n x\right)\,P_{n-1}(x) + \left(9\,x - 36\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	36							40
3	8	540							548
4	16	6372	9720						16108
5	32	71244	423792						495068
6	64	786564	12323016	7628256					20737900
7	128	8659116	303092928	664848000					976600172
8	256	95266404	6841179000	35647610112	11967264000				54551319772
9	512	1047967308	147112351632	1526345581248	1738069996032				3412575896732
10	1024	11527723332	3074290913448	57504133503072	148469321236608	31285259928576			240344533306060
11	2048	126805140972	63145810428768	1999643290241472	9782131819212288	6815302781160960			18660350506186508
12	4096	1394856956196	1283458641412824	65946027531027456	551688905251799424	848554348952028672	122675450060897280		1590149585294125948
13	8192	15343427402892	25920247080371760	2097252842763242880	28086349372919343360	79603655578874906112	37417781765683279872		147230975150748555068
14	16384	168777703348740	521442940233207816	65019412467154268640	1331535134359103650560	6269143255769861489664	6393993180137508602880	673520071782299037696	14733732666233863622380

Computed in your browser ?

Technical details

Node id: n-d3fd3793f8a4500d · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-cdacee00e0ff81b8

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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

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:   1;
  1:   2;
  2:   4,         36;
  3:   8,        540;
  4:  16,       6372,         9720;
  5:  32,      71244,       423792;
  6:  64,     786564,     12323016,       7628256;
  7: 128,    8659116,    303092928,     664848000;
  8: 256,   95266404,   6841179000,   35647610112,   11967264000;
  9: 512, 1047967308, 147112351632, 1526345581248, 1738069996032;

%F Formula

The row polynomials satisfy P(n,x) = (2+(18*n-18)*x)*P(n-1,x) + (9*x-36*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + (18*n - 18)*x)*P[n - 1, x] + (9*x - 36*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (2 + (18*n - 18)*x)*v[i - 1] + (9*x - 36*x^2)*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-a38dbd3442ab05b8
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-a38dbd3442ab05b8
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-a38dbd3442ab05b8

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

gamma_polynomial scale 1, shift 0 e-a38dbd3442ab05b8

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-56de0822e00a89e3 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

No candidate equalities or related sequences.

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