A142963 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 2\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-3 - 7\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	15	18	4						37
3	111	198	96	8					413
4	1071	2562	2004	528	16				6181
5	12663	37998	41136	18792	3024	32			113645
6	177255	639738	885060	577200	172320	17760	64		2469397
7	2866671	12093030	20385216	17326440	7642080	1579968	105600	128	61999133

All 15 rows

Technical details

Node id: n-c25403f40cbbc459 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-ff9c3889e2bd0aed

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:          3,          2;
  2:         15,         18,           4;
  3:        111,        198,          96,           8;
  4:       1071,       2562,        2004,         528,          16;
  5:      12663,      37998,       41136,       18792,        3024,         32;
  6:     177255,     639738,      885060,      577200,      172320,      17760,         64;
  7:    2866671,   12093030,    20385216,    17326440,     7642080,    1579968,     105600,       128;
  8:   52587711,  253921554,   505029492,   530034192,   311102640,   98768448,   14635392,    631296,     256;
  9: 1078805223, 5867608734, 13452958224, 16818013800, 12356943984, 5336410464, 1267053312, 137509632, 3782400, 512;

%F Formula

The row polynomials satisfy P(n,x) = (4*n-1+(4*n-2)*x)*P(n-1,x) - (3+7*x+4*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[(4*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (3 + 7*x + 4*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] = (4*n - 1 + (4*n - 2)*x)*v[i - 1] - (3 + 7*x + 4*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, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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