A028246 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in [−2, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	6	7	2													15
3	26	49	30	6												111
4	150	391	374	156	24											1095
5	1082	3601	4710	3030	960	120										13503
6	9366	37927	63062	55140	26760	6840	720									199815
7	94586	451249	911190	1009806	663600	258720	55440	5040								3449631
8	1091670	5995591	14250854	19150236	15916824	8381520	2731680	504000	40320							68062695
9	14174522	88073041	240899430	380739750	383251680	254850120	111978720	31358880	5080320	362880						1510769343
10	204495126	1418137447	4387780022	7976848980	9436983720	7592455080	4207699440	1586390400	389491200	56246400	3628800					37260156615
11	3245265146	24846302449	85799608950	176404548606	239949718800	226749048240	151918184880	72172734480	23830329600	5209142400	678585600	39916800				1010843385951
12	56183135190	470675213191	1794516768134	4117446611916	6332404740024	6877500632640	5409276269760	3104685460800	1290735008640	379109808000	74684332800	8861529600	479001600			29916558512295
13	1053716696762	9587626273681	40002164242950	101334700955670	173898459134400	213490849692120	192916125642240	129924459624960	65220570455040	24104486165760	6375950380800	1143177235200	124540416000	6227020800		959183053936383
14	21282685940886	209000505036967	947200772224982	2625875876316420	4974825728625480	6813830248144920	6958210045746960	5382204578150400	3169011259814400	1413534240518400	470225173017600	113135627404800	18612565171200	1874333260800	87178291200	33118910817665415

Technical details

Node id: n-5d7f56cbbf85a498 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d2b16266d15b8c0d

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:        2,        1;
  2:        6,        7,         2;
  3:       26,       49,        30,         6;
  4:      150,      391,       374,       156,        24;
  5:     1082,     3601,      4710,      3030,       960,       120;
  6:     9366,    37927,     63062,     55140,     26760,      6840,       720;
  7:    94586,   451249,    911190,   1009806,    663600,    258720,     55440,     5040;
  8:  1091670,  5995591,  14250854,  19150236,  15916824,   8381520,   2731680,   504000,   40320;
  9: 14174522, 88073041, 240899430, 380739750, 383251680, 254850120, 111978720, 31358880, 5080320, 362880;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (2+3*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[(2 + x)*P[n - 1, x] + (2 + 3*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] = (2 + x)*v[i - 1] + (2 + 3*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 [-2, -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 Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

Related ?

Same fingerprint ?

A130850 → x ↦ x/(x+1)

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741