A142461 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 12\,x + 12\,n x\right)\,P_{n-1}(x) + \left(6\,x - 24\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	12							13
3	1	108							109
4	1	792	1296						2089
5	1	5592	35856						41449
6	1	39204	667440	430272					1136917
7	1	274500	10558512	24193728					35026741
8	1	1921584	153730656	839787264	290324736				1285764241
9	1	13451184	2136852576	23335029504	27413012736				52898346001
10	1	94158396	28908982944	571576715136	1525386380544	328956152832			2454922389853
11	1	659108892	384855984288	12941404359552	65570341840128	46806913870848			125704175163709
12	1	4613762376	5074311556080	278214585511680	2415742807576320	3811546636240896	561682966450176		7072265921097529
13	1	32296336776	66519701714160	5773211034105600	80425020346248960	234113600487131136	112259389028438016		432637772893974649
14	1	226074357588	869019238738512	116875137433135680	2495575235521094400	12083022835876002816	12581687011588927488	1347112668341256192	28625142134073512677

Computed in your browser ?

Technical details

Node id: n-e216abf6fe8d06ac · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-1f4ee20ba4318b4a

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,      108;
  4: 1,      792,       1296;
  5: 1,     5592,      35856;
  6: 1,    39204,     667440,      430272;
  7: 1,   274500,   10558512,    24193728;
  8: 1,  1921584,  153730656,   839787264,   290324736;
  9: 1, 13451184, 2136852576, 23335029504, 27413012736;

%F Formula

The row polynomials satisfy P(n,x) = (1+(12*n-12)*x)*P(n-1,x) + (6*x-24*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[(1 + (12*n - 12)*x)*P[n - 1, x] + (6*x - 24*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] = (1 + (12*n - 12)*x)*v[i - 1] + (6*x - 24*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-de18761c144e2c93
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-de18761c144e2c93
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-de18761c144e2c93

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-de18761c144e2c93

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-f43e8fb782fe1ff0 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