A272867 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 8\,x + 19\,x^{2} + 18\,x^{3} + 6\,x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	6	6																											13
2	1	12	48	72	36																									169
3	1	18	126	432	756	648	216																							2197
4	1	24	240	1296	4104	7776	8640	5184	1296																					28561
5	1	30	390	2880	13320	40176	79920	103680	84240	38880	7776																			371293
6	1	36	576	5400	32940	137376	400896	824256	1185840	1166400	746496	279936	46656																	4826809
7	1	42	798	9072	68796	367416	1422792	4053888	8536752	13226976	14859936	11757312	6205248	1959552	279936															62748517
8	1	48	1056	14112	128016	834624	4040064	14795136	41396832	88770816	145442304	180278784	165908736	109734912	49268736	13436928	1679616													815730721
9	1	54	1350	20736	219024	1687392	9815904	44043264	154408032	425922624	926448192	1585557504	2120235264	2186860032	1703130624	967458816	377913600	90699264	10077696											10604499373
10	1	60	1680	29160	351540	3125952	21254400	113063040	477563040	1616630400	4408432128	9699782400	17192269440	24421616640	27545702400	24307402752	16401450240	8162933760	2821754880	604661760	60466176									137858491849
11	1	66	2046	39600	536580	5410152	42119352	259345152	1283467680	5160386880	16973592768	45850157568	101841556608	185773927680	277229018880	336111316992	327520081152	252416051712	150208058880	66512793600	20618966016	3990767616	362797056							1792160394037
12	1	72	2448	52272	786456	8867232	77799744	544522176	3092254704	14417263872	55636720128	178654035456	478784058624	1071924212736	2002921924608	3114128996352	4007562096384	4234204440576	3629824856064	2482257457152	1320944080896	526781325312	148021198848	26121388032	2176782336					23298085122481
13	1	78	2886	67392	1114776	13899600	135721872	1064524032	6826186224	36237925152	160693831584	598977939456	1884528592128	5016552777216	11307171552768	21563205820416	34709867622144	46964350996992	53080424077824	49666433236992	37993437960192	23345990553600	11234373636096	4074936532992	1047032303616	169789022208	13060694016			302875106592253
14	1	84	3360	85176	1536444	20992608	225808128	1962252864	14027661648	83582186688	419078499840	1780568479872	6442559218368	19917591966720	52713659768832	119505551800320	231932131861248	384602791652352	543125735792640	649935083685888	654474581849088	549305217736704	379270944718848	211557121671168	92902893318144	30901602041856	7313988648960	1097098297344	78364164096	3937376385699289

Technical details

Node id: n-63b66c53ded4a0a7 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-84fdcc1afa65c8bf

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,  6,    6;
  2: 1, 12,   48,    72,     36;
  3: 1, 18,  126,   432,    756,     648,     216;
  4: 1, 24,  240,  1296,   4104,    7776,    8640,     5184,      1296;
  5: 1, 30,  390,  2880,  13320,   40176,   79920,   103680,     84240,     38880,      7776;
  6: 1, 36,  576,  5400,  32940,  137376,  400896,   824256,   1185840,   1166400,    746496,     279936,      46656;
  7: 1, 42,  798,  9072,  68796,  367416, 1422792,  4053888,   8536752,  13226976,  14859936,   11757312,    6205248,    1959552,     279936;
  8: 1, 48, 1056, 14112, 128016,  834624, 4040064, 14795136,  41396832,  88770816, 145442304,  180278784,  165908736,  109734912,   49268736,  13436928,   1679616;
  9: 1, 54, 1350, 20736, 219024, 1687392, 9815904, 44043264, 154408032, 425922624, 926448192, 1585557504, 2120235264, 2186860032, 1703130624, 967458816, 377913600, 90699264, 10077696;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+8*x+19*x^2+18*x^3+6*x^4)*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 + 8*x + 19*x^2 + 18*x^3 + 6*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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 + 8*x + 19*x^2 + 18*x^3 + 6*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)

Related ?

Candidate equalities ?

A272867 → x ↦ x+1 → reverse 15 rows; no all-index equality certificate was found

Same fingerprint ?

A272867 → x ↦ x+1

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.