A257616 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	40	40	4													84
3	640	960	336	8												1944
4	14080	28160	16800	2720	16											61776
5	394240	985600	832000	262400	21824	32										2496096
6	13404160	40212480	43929600	20838400	3891840	174720	64									122451264
7	536166400	1876582400	2516828160	1600614400	479037440	56232960	1398016	128								7066859904
8	24663654400	98654617600	156882288640	125355704320	52154798080	10480476160	801241600	11184640	256							469003965696
9	1282510028800	5771295129600	10616094720000	10223620915200	5471716884480	1586595225600	222429388800	11329228800	89478144	512						35185680999936
10	74385581670400	371927908352000	776905113600000	876053004288000	574535968849920	219519606620160	46144922419200	4629855436800	159503984640	715827200	1024					2944262181049344
11	4760677226905600	26183724747980800	61227028774912000	79143693877248000	61527770942668800	29296033441382400	8316685239582720	1301624802508800	95149172490240	2240214056960	5726621696	2048				271854634166360064
12	333247405883392000	1999484435300352000	5174558603319705600	7544185693011968000	6787462157225164800	3879063259132723200	1396016568918343680	302579422136893440	35935526660014080	1938826874716160	31420263014400	45812981760	4096			27454503364539273216
13	25326802847137792000	164624218506395648000	467136692212885094400	758385403600515891200	775422430984378777600	517810017781363507200	226519446158573568000	63336329971551436800	10688736254860984320	976977884404121600	39279261702553600	440341812019200	366503870464	8192		3010266776172085272576
14	2076797833465298944000	14537584834257092608000	44881760624375234560000	80301960900909203456000	91994396020325954355200	70349777059756113920000	36310395412362559488000	12500308829839294464000	2770197571584228065280	369394940918866903040	26265568751963340800	792630703043379200	6168450406973440	2932030996480	16384	356119638398631291469824

Technical details

Node id: n-a513e6b009fe2695 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-72ea6b42576fa2d7

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,             2;
  2:            40,            40,              4;
  3:           640,           960,            336,              8;
  4:         14080,         28160,          16800,           2720,            16;
  5:        394240,        985600,         832000,         262400,         21824,            32;
  6:      13404160,      40212480,       43929600,       20838400,       3891840,        174720,           64;
  7:     536166400,    1876582400,     2516828160,     1600614400,     479037440,      56232960,      1398016,         128;
  8:   24663654400,   98654617600,   156882288640,   125355704320,   52154798080,   10480476160,    801241600,    11184640,      256;
  9: 1282510028800, 5771295129600, 10616094720000, 10223620915200, 5471716884480, 1586595225600, 222429388800, 11329228800, 89478144, 512;

%F Formula

The row polynomials satisfy P(n,x) = (6*n-2+(6*n-4)*x)*P(n-1,x) - (6*x+6*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[(6*n - 2 + (6*n - 4)*x)*P[n - 1, x] - (6*x + 6*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] = (6*n - 2 + (6*n - 4)*x)*v[i - 1] - (6*x + 6*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) = 6*x + 2.?
  - 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) = 6*x + 2.?
    
    ↑ 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