A257618 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 6\,x + 8\,n + 8\,n x\right)\,P_{n-1}(x) + \left(-8\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)

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

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	48	48	4													100
3	960	1440	496	8												2904
4	26880	53760	31872	4992	16											117520
5	967680	2419200	2035200	633600	49984	32										6105696
6	42577920	127733760	139238400	65587200	12004608	499968	64									387641920
7	2214051840	7749181440	10377400320	6570547200	1945359360	222082560	4999936	128								29083622784
8	132843110400	531372441600	844064686080	672390512640	277868666880	55020994560	4057485312	49999872	256							2517667897600
9	9033331507200	40649991782400	74710965288960	71788416860160	38243673538560	10988122521600	1511695564800	73634734080	499999744	512						247000331798016
10	686533194547200	3432665972736000	7165557374976000	8066233663488000	5272142478704640	2001806699397120	415923641548800	40776779366400	1331425210368	4999999488	1024					27082976229975040
11	57668788341964800	317178335880806400	741275183480832000	956900806557696000	741992961374945280	351686618708705280	99049132470435840	15283153634918400	1086824767733760	24025653780480	49999998976	2048				3282145880871819264
12	5305528527460761600	31833171164764569600	82345537486769356800	119923618423504896000	107675030915304652800	61323295932599500800	21942492764134440960	4710257094615367680	550058317083770880	28737957036687360	433061768036352	499999997952	4096			435638162145042042880
13	530552852746076160000	3448593542849495040000	9782010553717437235200	15866529074101459353600	16196249865580078694400	10785949796366077132800	4697760787825761976320	1304375950369798225920	217632897388861194240	19506645577875456000	755848118314598400	7801111824629760	4999999995904	8192		62849925620753059700736
14	57299708096576225280000	401097956676033576960000	1237907138141254739558400	2213169392059091936870400	2531970373860405647769600	1931967076205819068416000	993738333829706809344000	340297675113123702374400	74794645003599803842560	9842820931135686574080	684389696960574259200	19808073312672153600	140480012843286528	49999999991808	16384	9792789458217033286696960

Technical details

Node id: n-9f974a244ae6ecb1 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d6ac9abcd3bcb1be

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:            48,             48,              4;
  3:           960,           1440,            496,              8;
  4:         26880,          53760,          31872,           4992,             16;
  5:        967680,        2419200,        2035200,         633600,          49984,             32;
  6:      42577920,      127733760,      139238400,       65587200,       12004608,         499968,            64;
  7:    2214051840,     7749181440,    10377400320,     6570547200,     1945359360,      222082560,       4999936,         128;
  8:  132843110400,   531372441600,   844064686080,   672390512640,   277868666880,    55020994560,    4057485312,    49999872,       256;
  9: 9033331507200, 40649991782400, 74710965288960, 71788416860160, 38243673538560, 10988122521600, 1511695564800, 73634734080, 499999744, 512;

%F Formula

The row polynomials satisfy P(n,x) = (8*n-4+(8*n-6)*x)*P(n-1,x) - (8*x+8*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[(8*n - 4 + (8*n - 6)*x)*P[n - 1, x] - (8*x + 8*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] = (8*n - 4 + (8*n - 6)*x)*v[i - 1] - (8*x + 8*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) = 8*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) = 8*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