A257627 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 - 14\,x + 14\,n x\right)\,P_{n-1}(x) + \left(7\,x - 28\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	9	42							51
3	27	672							699
4	81	7854	9408						17343
5	243	83076	379848						463167
6	729	847770	9946608	5317872					16112979
7	2187	8538936	216567456	406133952					631242531
8	6561	85603686	4279372272	18843048000	5685875328				28893905847
9	19683	856771692	79940038248	691877999232	703867479168				1476542308023
10	59049	8570196978	1442944276032	22200874658928	50878767821952	9854144708352			84385301721291
11	177147	85710236640	25489914754080	654028311000960	2820490783380480	1799062997932032			5299157717481339
12	531441	857129647038	444128040636000	18194691109922880	133217196054862080	186825006823397376	25186881971048448		363868761130045263
13	1594323	8571385752468	7670174841397320	486414927189381120	5658087130934246400	14559513238361376768	6364509879752306688		27076203922466055087
14	4782969	85714147691466	131712965709634512	12640400282561209200	223069363923520988160	949327602223129565184	897902500600031625216	89103138316532293632	2172174804025637790339

Computed in your browser ?

Technical details

Node id: n-f8babbbcd1463e71 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-eca81e4e5309813a

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:     3;
  2:     9,        42;
  3:    27,       672;
  4:    81,      7854,        9408;
  5:   243,     83076,      379848;
  6:   729,    847770,     9946608,      5317872;
  7:  2187,   8538936,   216567456,    406133952;
  8:  6561,  85603686,  4279372272,  18843048000,   5685875328;
  9: 19683, 856771692, 79940038248, 691877999232, 703867479168;

%F Formula

The row polynomials satisfy P(n,x) = (3+(14*n-14)*x)*P(n-1,x) + (7*x-28*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[(3 + (14*n - 14)*x)*P[n - 1, x] + (7*x - 28*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] = (3 + (14*n - 14)*x)*v[i - 1] + (7*x - 28*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-c57a2725c54d20ba
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-c57a2725c54d20ba
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-c57a2725c54d20ba

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-c57a2725c54d20ba

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-4d5a49b3ac34613a 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