A290595 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 4\,x + 6\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	2							5
3	27	27	4						58
4	405	540	171	8					1124
5	8505	14175	6885	951	16				30532
6	229635	459270	300510	72198	4995	32			1066640
7	7577955	17681895	14543550	5038362	682479	25551	64		45549856
8	295540245	788107320	780529365	354760560	73848915	6073056	129099	128	2298988688

All 15 rows

Technical details

Node id: n-c2202905f6c6d864 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-29d82cb73104e710

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:            1;
   2:            3,             2;
   3:           27,            27,             4;
   4:          405,           540,           171,             8;
   5:         8505,         14175,          6885,           951,           16;
   6:       229635,        459270,        300510,         72198,         4995,           32;
   7:      7577955,      17681895,      14543550,       5038362,       682479,        25551,          64;
   8:    295540245,     788107320,     780529365,     354760560,     73848915,      6073056,      129099,       128;
   9:  13299311025,   39897933075,   46202791635,   26040379365,   7403662035,    994529745,    52070121,    648567,     256;
  10: 678264862275, 2260882874250, 2996778084300, 2017751766030, 731429268570, 139454339310, 12658141188, 436017978, 3249747, 512;

%F Formula

The row polynomials satisfy P(n,x) = (6*n-9+(3*n-4)*x)*P(n-1,x) - (3*x+3*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(6*n - 9 + (3*n - 4)*x)*P[n - 1, x] - (3*x + 3*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (6*n - 9 + (3*n - 4)*x)*v[i - 1] - (3*x + 3*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - 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