A257623 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 12\,x + 15\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-1}'(x)\)

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

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	66	66													141
3	27	582	1584	1056												3249
4	81	4818	26994	44352	22176											98421
5	243	39030	403920	1230240	1441440	576576										3691449
6	729	313698	5680290	28607040	58988160	53621568	17873856									165085341
7	2187	2513958	77294448	605811360	1957475520	3035096064	2252105856	643458816								8573758209
8	6561	20124786	1032481362	12141315648	57743975520	135877053312	168407471232	105527245824	26381811456							507131485701
9	19683	161037654	13643630352	235063383456	1583079065568	5305900856256	9769635203328	10072704305664	5461034971392	1213563326976						33654785800329
10	59049	1288419330	179138608770	4449438987840	41347149560640	189725279679936	486880888253760	734469630935040	647805380302080	309458648378880	61891729675776					2476208572861101
11	177147	10307708934	2342974526640	82956119521440	1044422658639360	6387333719614848	21948552982452096	45437557952666880	57968886221326080	44653062616081920	19062652740139008	3465936861843456				200053715154697809
12	531441	82462734354	30572053644594	1530697743812160	25763799658655520	206000333273839104	922309593728979456	2516775109569616896	4355601983576363520	4809915771752931840	3287626788647545344	1268532891434704896	211422148572450816			17605509772475809941
13	1594323	659705063478	398343787457616	28041712246932384	624712044769132320	6437868122792538432	36822226924545334272	128840599537888702464	290478664786137924096	431233648288653450240	419549924157719427072	257682007867475423232	90700101737581400064	13953861805781753856		1676352055989086133849
14	4782969	5277650073762	5185725992647266	511124321044104768	14957252986875623424	196502820602188516416	1414707400316564722368	6221542951723402727424	17773047166857367537152	34061233698017710119936	44229062043080455597056	38440352209145862463488	21432708889383629021184	6935069317473531666432	990724188210504523776	171710424253122784127421

Technical details

Node id: n-6a4d5848d20747a3 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-04577fbe25f76a46

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,        66,          66;
  3:    27,       582,        1584,         1056;
  4:    81,      4818,       26994,        44352,         22176;
  5:   243,     39030,      403920,      1230240,       1441440,        576576;
  6:   729,    313698,     5680290,     28607040,      58988160,      53621568,      17873856;
  7:  2187,   2513958,    77294448,    605811360,    1957475520,    3035096064,    2252105856,      643458816;
  8:  6561,  20124786,  1032481362,  12141315648,   57743975520,  135877053312,  168407471232,   105527245824,   26381811456;
  9: 19683, 161037654, 13643630352, 235063383456, 1583079065568, 5305900856256, 9769635203328, 10072704305664, 5461034971392, 1213563326976;

%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) + (5*x+15*x^2+15*x^3+5*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] + (5*x + 15*x^2 + 15*x^3 + 5*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] + (5*x + 15*x^2 + 15*x^3 + 5*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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.?
  - 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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257623 → x ↦ x+1

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