A257613 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 2\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

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

Technical details

Node id: n-6497b30746d46dc5 · revision 0 · JSON
Definition source: derived:shift_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:           8,            4;
  2:          80,           80,           16;
  3:         960,         1440,          608,           64;
  4:       13440,        26880,        17600,         4160,          256;
  5:      215040,       537600,       480000,       182400,        27008,        1024;
  6:     3870720,     11612160,     13171200,      6988800,      1729280,      170240,        4096;
  7:    77414400,    270950400,    373309440,    255897600,     90639360,    15536640,     1054208,      16384;
  8:  1703116800,   6812467200,  11054776320,   9320693760,   4339722240,  1092833280,   134835200,    6456320,    65536;
  9: 40874803200, 183936614400, 343719936000, 344648908800, 199925268480, 67376332800, 12546355200, 1143244800, 39262208, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+6+(2*n+2)*x)*P(n-1,x) - (2*x+2*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[(2*n + 6 + (2*n + 2)*x)*P[n - 1, x] - (2*x + 2*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] = (2*n + 6 + (2*n + 2)*x)*v[i - 1] - (2*x + 2*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) = 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