A257607 → γ

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 - 2\,x + 2\,n x\right)\,P_{n-1}(x) + \left(x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5								5
2	25	10							35
3	125	160							285
4	625	1710	320						2655
5	3125	15260	9080						27465
6	15625	122810	155120	18160					311715
7	78125	924360	2068320	765760					3836565
8	390625	6639910	23721840	18536000	1531520				50819895
9	1953125	46089460	245731800	338062720	87927680				719764785
10	9765625	311693010	2365375040	5161819760	2819725440	175855360			10844234235
11	48828125	2065470560	21544713440	69679058560	66672087040	13037455360			173047613085
12	244140625	13467042110	187991464160	859058456640	1296839368960	530407075840	26074910720		2914082459055
13	1220703125	86661627660	1585281091320	9880331079680	21980255800320	15678785710080	2408452321280		51620988333465
14	6103515625	551708047210	13003523447760	107577708281200	336146937318400	376590415104000	120565689794560	4816904642560	959258990151315

Computed in your browser ?

Technical details

Node id: n-37abe93920806e29 · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-4d9935dd32a36549

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:       5;
  2:      25,       10;
  3:     125,      160;
  4:     625,     1710,       320;
  5:    3125,    15260,      9080;
  6:   15625,   122810,    155120,     18160;
  7:   78125,   924360,   2068320,    765760;
  8:  390625,  6639910,  23721840,  18536000,  1531520;
  9: 1953125, 46089460, 245731800, 338062720, 87927680;

%F Formula

The row polynomials satisfy P(n,x) = (5+(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[(5 + (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] = (5 + (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-a5ce40235f8a7dd2
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-a5ce40235f8a7dd2
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-a5ce40235f8a7dd2

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-a5ce40235f8a7dd2

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 + 5.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-5eeeda74dfe3072e 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