A144441 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 8\,x + 8\,n x\right)\,P_{n-1}(x) + \left(4\,x - 16\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	12							13
3	1	80							81
4	1	428	688						1117
5	1	2176	13360						15537
6	1	10924	174176	109632					294733
7	1	54672	1925856	4265472					6246001
8	1	273420	19563280	102368384	34562304				156767389
9	1	1367168	189412368	1964517376	2242515200				4397812113
10	1	6835916	1782366400	33193473728	85680648960	18078370816			138741695821
11	1	34179664	16484264896	517826395136	2526620261120	1759506231296			4820471332113
12	1	170898412	150846663536	7661991436544	63798374139392	97931239719936	14148363333632		183690986191453
13	1	854492160	1371428563440	109326012200960	1454419254904320	4108210487623680	1927646943784960		7600974981569521
14	1	4272460908	12418735974688	1520584401834304	30878031982373376	144704383932827648	147179950256472064	15477769003614208	339773142585557197

Computed in your browser ?

Technical details

Node id: n-273eab41f1e8cbee · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-a37ca93b409c040e

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: 1;
  2: 1,      12;
  3: 1,      80;
  4: 1,     428,       688;
  5: 1,    2176,     13360;
  6: 1,   10924,    174176,     109632;
  7: 1,   54672,   1925856,    4265472;
  8: 1,  273420,  19563280,  102368384,   34562304;
  9: 1, 1367168, 189412368, 1964517376, 2242515200;

%F Formula

The row polynomials satisfy P(n,x) = (1+(8*n-8)*x)*P(n-1,x) + (4*x-16*x^2)*d/dx P(n-1,x) + 4*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (8*n - 8)*x)*P[n - 1, x] + (4*x - 16*x^2)*D[P[n - 1, x], x] + 4*x*P[n - 2, 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + (8*n - 8)*x)*v[i - 1] + (4*x - 16*x^2)*deriv(v[i - 1], x) + 4*x*v[i - 2]); 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-24cd925870a5c300
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-24cd925870a5c300
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-24cd925870a5c300

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-24cd925870a5c300

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n,…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-494af060e65fc342 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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