A144699 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 4\,x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	5														11
2	42	66	25													133
3	336	756	546	125												1763
4	3024	8736	9114	4026	625											25525
5	30240	105840	142800	91980	27906	3125										401891
6	332640	1360800	2237760	1878240	839202	186186	15625									6850453
7	3991680	18627840	35985600	37044000	21739536	7177716	1210866	78125								125855363
8	51891840	271434240	600747840	730296000	529051824	231338016	58720074	7733946	390625							2481604405
9	726485760	4203239040	10466184960	14641482240	12593478240	6843508560	2320464720	465178140	48747426	1953125						52310722211
10	10897286400	69016147200	190702512000	301292006400	299584565280	194369868000	82475759520	22285142880	3597478962	304203306	9765625					1174534735573
11	174356582400	1198701504000	3636061228800	6395928739200	7210518860160	5433498766080	2768456275200	942848121600	207060974736	27311775876	1883813586	48828125				27996675469763
12	2964061900800	21968929382400	72521440992000	140447859552000	176883818791680	151728211514880	90221505454080	37113043968000	10349181867024	1874982004896	204369126234	11595850266	244140625			706289244544885
13	53353114214400	423860851814400	1511845925990400	3194539508160000	4442643496811520	4271956393985280	2903396603627520	1400680043642880	474622258350240	110017474847280	16634809049040	1511754835500	71039945346	1220703125		18805134495976931
14	1013709170073600	8589851388518400	32907015222681600	75302166946406400	114557745074572800	122004398471155200	93279588968966400	51648919767244800	20628325321924320	5846397331975200	1139870555066880	145172511076320	11079563465922	433563890826	6103515625	527074679960534293

Technical details

Node id: n-3591d66c31ac5434 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-43aff1813432376c

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:         6,          5;
  2:        42,         66,          25;
  3:       336,        756,         546,         125;
  4:      3024,       8736,        9114,        4026,         625;
  5:     30240,     105840,      142800,       91980,       27906,       3125;
  6:    332640,    1360800,     2237760,     1878240,      839202,     186186,      15625;
  7:   3991680,   18627840,    35985600,    37044000,    21739536,    7177716,    1210866,     78125;
  8:  51891840,  271434240,   600747840,   730296000,   529051824,  231338016,   58720074,   7733946,   390625;
  9: 726485760, 4203239040, 10466184960, 14641482240, 12593478240, 6843508560, 2320464720, 465178140, 48747426, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (n+5+(n+4)*x)*P(n-1,x) - (x+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[(n + 5 + (n + 4)*x)*P[n - 1, x] - (x + 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] = (n + 5 + (n + 4)*x)*v[i - 1] - (x + 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 of 5-Eulerian numbers.?
  - 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