A391533 → reverse → 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

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

\(P_{0}(x) = 1 + 6\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	6	9															16
1	1	8	21	18														48
2	1	10	37	60	36													144
3	1	12	57	134	156	72												432
4	1	14	81	248	424	384	144											1296
5	1	16	109	410	920	1232	912	288										3888
6	1	18	141	628	1740	3072	3376	2112	576									11664
7	1	20	177	910	2996	6552	9520	8864	4800	1152								34992
8	1	22	217	1264	4816	12544	22624	27904	22528	10752	2304							104976
9	1	24	261	1698	7344	22176	47712	73152	78336	55808	23808	4608						314928
10	1	26	309	2220	10740	36864	92064	168576	224640	212480	135424	52224	9216					944784
11	1	28	361	2838	15180	58344	165792	352704	561792	661760	560384	323072	113664	18432				2834352
12	1	30	417	3560	20856	88704	282480	684288	1267200	1785344	1883904	1443840	759808	245760	36864			8503056
13	1	32	477	4394	27976	130416	459888	1249248	2635776	4319744	5454592	5211648	3647488	1765376	528384	73728		25509168
14	1	34	541	5348	36764	186368	720720	2169024	5134272	9591296	14094080	16120832	14070784	9060352	4059136	1130496	147456	76527504

Technical details

Node id: n-5cb613a96f187602 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-063dae614330e0ae

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1,  6,   9;
  1: 1,  8,  21,   18;
  2: 1, 10,  37,   60,   36;
  3: 1, 12,  57,  134,  156,    72;
  4: 1, 14,  81,  248,  424,   384,   144;
  5: 1, 16, 109,  410,  920,  1232,   912,   288;
  6: 1, 18, 141,  628, 1740,  3072,  3376,  2112,   576;
  7: 1, 20, 177,  910, 2996,  6552,  9520,  8864,  4800,  1152;
  8: 1, 22, 217, 1264, 4816, 12544, 22624, 27904, 22528, 10752,  2304;
  9: 1, 24, 261, 1698, 7344, 22176, 47712, 73152, 78336, 55808, 23808, 4608;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1+3*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 6*x + 9*x^2.

%t Mathematica

P[0, x_] := 1 + 6*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 3*x + 2*x^2)*P[n - 1, 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 + 6*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 3*x + 2*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 - 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 reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    no recorded property covers this requirement
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Candidate equalities ?

A391533 → x ↦ x+1 → reverse 15 rows; no all-index equality certificate was found

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.