A142460 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

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

Technical details

Node id: n-59a56c7d8834e636 · revision 0 · JSON
Definition source: derived:shift_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:           2,            1;
  2:          14,           14,            1;
  3:         168,          252,           86,            1;
  4:        2856,         5712,         3374,          518,            1;
  5:       62832,       157080,       131880,        40740,         3110,            1;
  6:     1696464,      5089392,      5540640,      2598960,       469910,        18662,           1;
  7:    54286848,    190003968,    254218272,    160535760,     47222280,      5299644,      111974,         1;
  8:  2008613376,   8034453504,  12754016352,  10141461792,   4174029720,    819152208,    59079902,    671846,       1;
  9: 84361761792, 379627928064, 697368849408, 669193975296, 355506660432, 101580211656, 13815394152, 654581844, 4031078, 1;

%F Formula

The row polynomials satisfy P(n,x) = (5*n-3+(5*n-4)*x)*P(n-1,x) - (5*x+5*x^2)*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[(5*n - 3 + (5*n - 4)*x)*P[n - 1, x] - (5*x + 5*x^2)*D[P[n - 1, x], x]];
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] = (5*n - 3 + (5*n - 4)*x)*v[i - 1] - (5*x + 5*x^2)*deriv(v[i - 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]
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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/(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