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	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

All 15 rows

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