A142460 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	14	14													29
3	1	86	252	168												507
4	1	518	3374	5712	2856											12461
5	1	3110	40740	131880	157080	62832										395643
6	1	18662	469910	2598960	5540640	5089392	1696464									15414029
7	1	111974	5299644	47222280	160535760	254218272	190003968	54286848								711678747
8	1	671846	59079902	819152208	4174029720	10141461792	12754016352	8034453504	2008613376							37991478701
9	1	4031078	654581844	13815394152	101580211656	355506660432	669193975296	697368849408	379627928064	84361761792						2302113393723
10	1	24186470	7228617830	228901288560	2368046145360	11477937827664	30343693065840	46519485788160	41367392478720	19825014021120	3965002804224					156102726223949
11	1	145118822	79684101420	3749164030920	53620290958080	350523398717184	1250558806387968	2645699666480640	3417284065789440	2649381129077760	1133990802008064	206180145819648				11711067298489947
12	1	870712934	877540947374	60942833711760	1189761898645320	10293254767724544	48231454763390976	135263069797718016	237999534357150720	265397462700733440	182354443969065984	70513609870319616	11752268311719936			963056681681840621
13	1	5224277606	9659045411652	985615830756648	26021028044651640	293799385731035184	1773092976393682944	6412877065146359808	14762714491158746112	22204263727234068480	21773757389356836864	13431193239129329664	4736164129623134208	728640635326636032		86143519347244926843
14	1	31345665638	106286069471414	15885761837046192	563197058060547456	8211246645989250864	62898465682942121232	287602549589866807296	842547745547923903488	1641430140081434486784	2154062022037199898624	1884382205637797978112	1054484026537382977536	341732457968192299008	48818922566884614144	8326748971432927067789

Technical details

Node id: n-861b0207a722f305 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-b94fba28f209154b

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: 1,       2;
  2: 1,      14,        14;
  3: 1,      86,       252,         168;
  4: 1,     518,      3374,        5712,         2856;
  5: 1,    3110,     40740,      131880,       157080,        62832;
  6: 1,   18662,    469910,     2598960,      5540640,      5089392,      1696464;
  7: 1,  111974,   5299644,    47222280,    160535760,    254218272,    190003968,     54286848;
  8: 1,  671846,  59079902,   819152208,   4174029720,  10141461792,  12754016352,   8034453504,   2008613376;
  9: 1, 4031078, 654581844, 13815394152, 101580211656, 355506660432, 669193975296, 697368849408, 379627928064, 84361761792;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*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[((1 + 4*x + 5*x^2 + 2*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] = ((1 + 4*x + 5*x^2 + 2*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) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-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) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142460 → 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