A161198 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-3 - 4\,x - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	3														4
3	3	14	15													32
4	15	91	173	105												384
5	105	772	2030	2292	945											6144
6	945	8103	26762	42958	33717	10395										122880
7	10395	101418	399721	820444	929341	552666	135135									2949120
8	135135	1474359	6717643	16661587	24388093	21124773	10046745	2027025								82575360
9	2027025	24412680	125828748	364123736	649068374	731469176	509822316	201200040	34459425							2642411520
10	34459425	453529035	2602929636	8580849724	17952513342	24767275098	22564893716	13107024684	4408610985	654729075						95126814720
11	654729075	9340699590	58979772819	217697667112	521295597702	847581007044	948845757662	722896237032	359011127079	105020683110	13749310575					3805072588800
12	13749310575	211213460115	1453411319769	5928185784189	15954253895318	29788999895070	39420124072914	37004273403898	24165847120395	10462690268127	2704211233605	316234143225				167423193907200
13	316234143225	5201642347020	38708796857562	172683556030572	515152484197039	1084003344969560	1651387851053772	1836601390112504	1480921318866535	844788054176796	323764115076090	74878666134300	7905853580625			8036313307545600
14	7905853580625	138579380542575	1108164264808590	5362226415918474	17541268117751419	41009200697559053	70552786590522420	90502506731264444	86611270504700983	61104576963816345	30903380339675742	10613597760411930	2219370325141725	213458046676875		417888291992371200
15	213458046676875	3970913028487650	33939237185566605	176916876909247908	629118805240924059	1615945194248885582	3094192058173317877	4489598492869290168	4963076998833595417	4161550422659369822	2606424001121919039	1182765169381718628	367717333830772545	70125106689385650	6190283353629375	23401744351572787200

Technical details

Node id: n-4b3d4c96cc07b501 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-438412a981173c5d

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 -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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        1;
   2:        1,         3;
   3:        3,        14,         15;
   4:       15,        91,        173,        105;
   5:      105,       772,       2030,       2292,         945;
   6:      945,      8103,      26762,      42958,       33717,       10395;
   7:    10395,    101418,     399721,     820444,      929341,      552666,      135135;
   8:   135135,   1474359,    6717643,   16661587,    24388093,    21124773,    10046745,     2027025;
   9:  2027025,  24412680,  125828748,  364123736,   649068374,   731469176,   509822316,   201200040,   34459425;
  10: 34459425, 453529035, 2602929636, 8580849724, 17952513342, 24767275098, 22564893716, 13107024684, 4408610985, 654729075;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2*n-3+(4*n-4)*x+(2*n-1)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 3 + (4*n - 4)*x + (2*n - 1)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 3 + (4*n - 4)*x + (2*n - 1)*x^2)*v[i - 1])/(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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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.