A257606 → γ

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4								4
2	16	8							24
3	64	104							168
4	256	904	208						1368
5	1024	6568	4864						12456
6	4096	43080	68592	9728					125496
7	16384	264552	756192	342464					1379592
8	65536	1552136	7182672	6934400	684928				16419672
9	262144	8809256	61721664	106002176	33217024				210012264
10	1048576	48764872	493659568	1359231872	901749248	66434048			2870888184
11	4194304	264795880	3742195360	15438537920	18087848960	4204903424			41742475848
12	16777216	1416254088	27219498000	160460500480	299088170880	146371224576	8409806848		642982232088
13	67108864	7483923624	191642069760	1558735471360	4318231372800	3710046388224	669582966784		10455789301416
14	268435456	39164448584	1314498738288	14360705555200	56368147581440	76572731222016	28956107997184	1339165933568	178950789911736

Computed in your browser ?

Technical details

Node id: n-91ed3144de9cf292 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-d652dcc1240a5990

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:      4;
  2:     16,       8;
  3:     64,     104;
  4:    256,     904,      208;
  5:   1024,    6568,     4864;
  6:   4096,   43080,    68592,      9728;
  7:  16384,  264552,   756192,    342464;
  8:  65536, 1552136,  7182672,   6934400,   684928;
  9: 262144, 8809256, 61721664, 106002176, 33217024;

%F Formula

The row polynomials satisfy P(n,x) = (4+(2*n-2)*x)*P(n-1,x) + (x-4*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[(4 + (2*n - 2)*x)*P[n - 1, x] + (x - 4*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] = (4 + (2*n - 2)*x)*v[i - 1] + (x - 4*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-9a33444ab708fa7d
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-9a33444ab708fa7d
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-9a33444ab708fa7d

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-9a33444ab708fa7d

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) = x + 4.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-de9fa36753ad8047 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