A257625 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 12\,x + 15\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(6\,x + 18\,x^{2} + 18\,x^{3} + 6\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	9	72	72													153
3	27	702	1944	1296												3969
4	81	6480	37584	62208	31104											137457
5	243	58806	641520	1982880	2332800	933120										5949369
6	729	530712	10328472	53187840	110574720	100776960	33592320									308991753
7	2187	4780782	161295624	1302857136	4262025600	6642881280	4938071040	1410877440								18722791089
8	6561	43040160	2476803744	30263321088	146343262464	347075850240	431728496640	270888468480	67722117120							1296541366497
9	19683	387400806	37668538080	680112210240	4677587792640	15843800931840	29332141977600	30322577940480	16456474460160	3656994324480						101007745596009
10	59049	3486725352	569676880872	14960390100480	142617563447040	663173064529920	1714330370672640	2596465970380800	2294763938611200	1097098297344000	219419659468800					8743402418220153
11	177147	31380882462	8586993917304	324422375965776	4209723575481600	26163238032898560	90733114779310080	188842844235386880	241663327447449600	186451855633612800	79649336387174400	14481697524940800				832528178367197409
12	531441	282429005040	129181479349104	6967435785792768	121448673561181824	989678562350100480	4480468045577441280	12308718811323432960	21389286223118499840	23677575453278208000	16205019530408755200	6256093330774425600	1042682221795737600			86478067751882461137
13	1594323	2541864234006	1941111338297040	148641418129932000	3446332645010935680	36302852764388770560	210366682022123827200	742072004423807016960	1681672100322568273920	2504643256885256847360	2441570757612444057600	1501201728830413209600	528639886450438963200	81329213300067532800		9731395400327853492249
14	4782969	22876787671992	29147172445263672	3156409784817918720	96618375450413631360	1301384120575157498880	9511203298380825000960	42228640844000516505600	121384733328793718784000	233574959059878327091200	304097553142699386470400	264726589291719819264000	147734515959572673331200	47821577420439709286400	6831653917205672755200	1179312614338550275256553

Technical details

Node id: n-b29e1efbc9cdcce1 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-2a2e8db5a042c827

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:     3,         6;
  2:     9,        72,          72;
  3:    27,       702,        1944,         1296;
  4:    81,      6480,       37584,        62208,         31104;
  5:   243,     58806,      641520,      1982880,       2332800,         933120;
  6:   729,    530712,    10328472,     53187840,     110574720,      100776960,       33592320;
  7:  2187,   4780782,   161295624,   1302857136,    4262025600,     6642881280,     4938071040,     1410877440;
  8:  6561,  43040160,  2476803744,  30263321088,  146343262464,   347075850240,   431728496640,   270888468480,    67722117120;
  9: 19683, 387400806, 37668538080, 680112210240, 4677587792640, 15843800931840, 29332141977600, 30322577940480, 16456474460160, 3656994324480;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+12*x+15*x^2+6*x^3)*P(n-1,x) + (6*x+18*x^2+18*x^3+6*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[((3 + 12*x + 15*x^2 + 6*x^3)*P[n - 1, x] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*D[P[n - 1, x], 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] = ((3 + 12*x + 15*x^2 + 6*x^3)*v[i - 1] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*deriv(v[i - 1], x))/(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) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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