A256890 → γ

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 4 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	4	4								8
2	8	28								36
3	16	132	56							204
4	32	524	752							1308
5	64	1892	6152	1504						9612
6	128	6444	39744	32128						78444
7	256	21124	223416	399104	64256					708156
8	512	67468	1147152	3782848	1981952					6979932
9	1024	211620	5533160	30385760	34588800	3963904				74684268
10	2048	655340	25518560	218326720	450618880	166102528				861224076
11	4096	2011076	113870360	1448893440	4886980480	3866430976	332205056			10650395484
12	8192	6131532	495702960	9066392960	46708604160	66160860672	18123364352			140561064828
13	16384	18607588	2117705544	54254618080	407181126400	930212066304	541952078848	36246728704		1971982947852
14	32768	56281516	8917404288	313627201280	3311160647680	11397657980928	11777313161216	2494028873728		29302761583404
15	65536	169827588	37132936568	1764318900736	25512253509120	126139854934016	208195085099008	93110138830848	4988057747456	459747011850876

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Technical details

Node id: n-dedd8998f219c10f · revision 1 · JSON
Definition source: derived:gamma_polynomial
Fingerprint: f6-e3b00db85661a6a1

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    4,      4;
   2:    8,     28;
   3:   16,    132,       56;
   4:   32,    524,      752;
   5:   64,   1892,     6152,      1504;
   6:  128,   6444,    39744,     32128;
   7:  256,  21124,   223416,    399104,     64256;
   8:  512,  67468,  1147152,   3782848,   1981952;
   9: 1024, 211620,  5533160,  30385760,  34588800,   3963904;
  10: 2048, 655340, 25518560, 218326720, 450618880, 166102528;

%F Formula

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

%t Mathematica

P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*n*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 4 + 4*x; for(n = 2, N, my(i = n); v[i] = (2 + 2*n*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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-0784867e60003b7b
Coefficients of one sign. Every nonzero row, n >= 1: 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-0784867e60003b7b
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-0784867e60003b7b

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-0784867e60003b7b

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
- premise (sequence): Triangle 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 + 2.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-072c1088d2b33d0a root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

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