A144697 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	20	28	9													57
3	120	240	148	27												535
4	840	2160	1940	700	81											5721
5	6720	21000	24600	13200	3124	243										68887
6	60480	221760	319200	226800	81620	13468	729									924057
7	604800	2540160	4327680	3822000	1850520	475440	56788	2187								13679575
8	6652800	31449600	61810560	65197440	39707640	13955760	2661140	235900	6561							221677401
9	79833600	419126400	932601600	1143072000	839240640	375643800	99701400	14485200	969844	19683						3904694167
10	1037836800	5987520000	14869008000	20756736000	17840813760	9719075520	3327416400	685119600	77275220	3958108	59049					74304818457
11	14529715200	91329638400	250278336000	391883184000	385975497600	248039003520	104371935360	28087752000	4574368920	406166640	16068628	177147				1519491843415
12	217945728000	1482030950400	4440903667200	7705938240000	8556445497600	6340145011200	3171698449920	1059677498880	228634846440	29883213360	2111176340	64983100	531441			33235479793881
13	3487131648000	25499650176000	82921084646400	157908944793600	195148604851200	163862495596800	95118434611200	38082383539200	10306498762560	1809626418600	191966338200	10880797200	262058164	1594323		774347965831447
14	59281238016000	463788509184000	1626311022336000	3371620412160000	4589748145382400	4308133205376000	2848790232288000	1332008181504000	435499940715840	96929001449280	14011149297600	1217082812400	55714276820	1054609948	4782969	19147394894191257

Technical details

Node id: n-c781d765aa7ea874 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-5bb7eea759246da4

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,         3;
  2:       20,        28,         9;
  3:      120,       240,       148,         27;
  4:      840,      2160,      1940,        700,        81;
  5:     6720,     21000,     24600,      13200,      3124,       243;
  6:    60480,    221760,    319200,     226800,     81620,     13468,      729;
  7:   604800,   2540160,   4327680,    3822000,   1850520,    475440,    56788,     2187;
  8:  6652800,  31449600,  61810560,   65197440,  39707640,  13955760,  2661140,   235900,   6561;
  9: 79833600, 419126400, 932601600, 1143072000, 839240640, 375643800, 99701400, 14485200, 969844, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (n+3+(n+2)*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 + 3 + (n + 2)*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 + 3 + (n + 2)*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 3-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