A282629 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 8\,x + 10\,x^{2} + 3\,x^{3} - 3\,n x - 6\,n x^{2} - 3\,n x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	17	25													43
3	1	66	237	199												503
4	1	259	1716	3253	1876											7105
5	1	1028	11482	37063	46622	20257										116453
6	1	4101	75219	369776	762189	707820	245017									2164123
7	1	16390	493707	3485807	10484933	15619026	11407309	3266914								44774087
8	1	65543	3275638	32094515	132384331	281469263	325421812	194896823	47450923							1017058849
9	1	262152	22011840	293363643	1597845789	4551627807	7409095086	6961255731	3521472309	743935375						25100869733
10	1	1048585	149626269	2685214542	18842928402	69143345127	148666514448	194846328516	153618810840	67099463839	12497579698					667550860267
11	1	4194314	1026412153	24729065859	219885322434	1012218159846	2762064443784	4736134194927	5174013353880	3504781869328	1344497093090	223619318215				19002973427831
12	1	16777227	7088415816	229715822569	2558776339062	14502566364750	48873850820628	105309476517654	149489111724291	139606535725780	82732836434070	28253775489132	4240423494685			575804173925665
13	1	67108876	49182702774	2154836260879	29835418417360	205368523914141	838187400724944	2207749993011150	3910593243512511	4721166640145686	3842192374731838	2020688965252428	621127017329413	84855613320004		18483969276432005
14	1	268435469	342332761975	20414744525452	349605193799195	2893061692208926	14093161696934511	44479842673030860	95617145623667001	143290878215541151	150151231126767347	108104234505130270	51043326191427007	14251790382473873	1785410320771933	626080444967474971

Technical details

Node id: n-4e3b9877e6d1f6d9 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-80f00423ae1fa6cc

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,      4;
  2: 1,     17,       25;
  3: 1,     66,      237,       199;
  4: 1,    259,     1716,      3253,       1876;
  5: 1,   1028,    11482,     37063,      46622,      20257;
  6: 1,   4101,    75219,    369776,     762189,     707820,     245017;
  7: 1,  16390,   493707,   3485807,   10484933,   15619026,   11407309,    3266914;
  8: 1,  65543,  3275638,  32094515,  132384331,  281469263,  325421812,  194896823,   47450923;
  9: 1, 262152, 22011840, 293363643, 1597845789, 4551627807, 7409095086, 6961255731, 3521472309, 743935375;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1-(3*n-8)*x-(6*n-10)*x^2-(3*n-3)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*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[Together[((1 - (3*n - 8)*x - (6*n - 10)*x^2 - (3*n - 3)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + 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 - (3*n - 8)*x - (6*n - 10)*x^2 - (3*n - 3)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.