A257613 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 20\,x^{2} + 8\,x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	8														12
2	16	80	80													176
3	64	608	1440	960												3072
4	256	4160	17600	26880	13440											62336
5	1024	27008	182400	480000	537600	215040										1443072
6	4096	170240	1729280	6988800	13171200	11612160	3870720									37546496
7	16384	1054208	15536640	90639360	255897600	373309440	270950400	77414400								1084818432
8	65536	6456320	134835200	1092833280	4339722240	9320693760	11054776320	6812467200	1703116800							34464966656
9	262144	39262208	1143244800	12546355200	67376332800	199925268480	344648908800	343719936000	183936614400	40874803200						1194210988032
10	1048576	237670400	9538580480	139182489600	984164966400	3876975083520	9113037373440	13079937024000	11240570880000	5313724416000	1062744883200					44820114415616
11	4194304	1434411008	78685347840	1506287861760	13758534451200	70024290631680	215594149478400	417699613900800	512570032128000	386675638272000	163662712012800	29756856729600				1811328239419392
12	16777216	8640020480	643826892800	16007102791680	186190443479040	1200476620062720	4709943623024640	11830476039782400	19440792148377600	20808544813056000	13981471683379200	5356234211328000	892705701888000			78423494870859776
13	67108864	51974340608	5237015347200	167796950630400	2458384760832000	19785719776542720	96967677129523200	307147441176576000	649086315842764800	924366997448294400	876577485540556800	530743296629145600	185682785992704000	28566582460416000		3621555772764782592
14	268435456	312382914560	42415866183680	1740813690470400	31849774438809600	316334233044910080	1907625790050140160	7468007483768832000	19738970022739968000	35914145524088832000	45071401586629017600	38343495307493376000	21121416906670080000	6798846625579008000	971263803654144000	177685140600365121536

Technical details

Node id: n-96cfdd1a383cbf1a · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-cef044b741d89456

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:      4,        8;
  2:     16,       80,         80;
  3:     64,      608,       1440,         960;
  4:    256,     4160,      17600,       26880,       13440;
  5:   1024,    27008,     182400,      480000,      537600,       215040;
  6:   4096,   170240,    1729280,     6988800,    13171200,     11612160,      3870720;
  7:  16384,  1054208,   15536640,    90639360,   255897600,    373309440,    270950400,     77414400;
  8:  65536,  6456320,  134835200,  1092833280,  4339722240,   9320693760,  11054776320,   6812467200,   1703116800;
  9: 262144, 39262208, 1143244800, 12546355200, 67376332800, 199925268480, 344648908800, 343719936000, 183936614400, 40874803200;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4+16*x+20*x^2+8*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*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[((4 + 16*x + 20*x^2 + 8*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*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] = ((4 + 16*x + 20*x^2 + 8*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.?
    
    ↑ 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