A167884 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 16\,x + 16\,n x\right)\,P_{n-1}(x) + \left(8\,x - 32\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	16							17
3	1	176							177
4	1	1632	2816						4449
5	1	14752	100096						114849
6	1	132848	2409728	1601536					4144113
7	1	1195728	49467648	117149696					167813073
8	1	10761664	936608256	5303189504	1874395136				8124954561
9	1	96855104	16955460096	192522665984	231557103616				441132084801
10	1	871696080	299090593280	6169503457280	16882472386560	3704913657856			27056851791057
11	1	7845264880	5196117184000	182950283386880	951969810022400	692140576342016			1832264632200177
12	1	70607384096	89463710270720	5155722209280000	46051026401689600	74072314511097856	11074249221472256		136442846661194529
13	1	635466457056	1532180256057600	140344410146652160	2014633203346636800	5984230584663146496	2912952276207271936		11053693290086222049
14	1	5719198113712	26158906449421056	3729244210538598400	82201469646864056320	406524110238919950336	429977729597987356672	46607236419316350976	969065954739273847473

Computed in your browser ?

Technical details

Node id: n-a19d1a6e6c32556d · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-37e18a3c0000c110

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,       16;
  3: 1,      176;
  4: 1,     1632,        2816;
  5: 1,    14752,      100096;
  6: 1,   132848,     2409728,      1601536;
  7: 1,  1195728,    49467648,    117149696;
  8: 1, 10761664,   936608256,   5303189504,   1874395136;
  9: 1, 96855104, 16955460096, 192522665984, 231557103616;

%F Formula

The row polynomials satisfy P(n,x) = (1+(16*n-16)*x)*P(n-1,x) + (8*x-32*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 + (16*n - 16)*x)*P[n - 1, x] + (8*x - 32*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 + (16*n - 16)*x)*v[i - 1] + (8*x - 32*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-15b23c3dbd64ff2a
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-15b23c3dbd64ff2a
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-15b23c3dbd64ff2a

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-15b23c3dbd64ff2a

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+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 e-46d699555fa04487 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