A144443 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 6\,x\right)\,P_{n-2}(x) + \left(-4 - 5\,x + 6\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + x\)

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	22	22	1													45
3	320	480	162	1												963
4	6532	13064	7674	1142	1											28413
5	171752	429380	359504	109876	8002	1										1078515
6	5535256	16605768	18049008	8421736	1499262	56022	1									50167053
7	211370240	739795840	988786320	622476200	181709920	19986600	392162	1								2764517283
8	9333502096	37334008384	59221536976	46995581584	19255629388	3741632584	263299234	2745142	1							176147935389
9	467943326240	2105744968080	3866152258560	3704723053920	1962334509912	557433565500	74899480944	3447204156	19216002	1						12742697583315
10	26260827282016	131304136410080	273871536084720	307661325878400	200266964578056	75463626082104	15457858347288	1472947262376	44983852902	134512022	1					1031804340289965
11	1630978951442432	8970384232933376	20951135690150240	27002228858675760	20873584012120320	9839832332467872	2745070723853376	416375484040320	28636570446720	585981479920	941584162	1				92458813779194499
12	111064133661777472	666384801970664832	1722888090892318176	2505913103063829920	2244594480351145056	1273132124698913856	452461588371195648	96024539858160960	10991226096634068	552569289396104	7626098984394	6591089142	1			9084014290944109629
13	8228531764680187520	53485456470421218880	151646372486896772352	245715027503298840256	250342199563292674880	166187223522025846560	72013731138496657920	19832832722442751872	3265202686442397768	286007373515333620	10609103422691984	99197665015236	46137624002	1		971013293578738012851
14	658948925976385666432	4612642481834699665024	14230727288806907396224	25420011468990348732032	29036282898746108898752	22099288511712689799872	11322131684951540086944	3853528999177557001344	838943391292267318008	108726737763429185960	7365727578667463168	203007528481802216	1289978292724942	322963368022	1	112188802414660339108941

Technical details

Node id: n-9109503bbc9c6251 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4686d3e0771fa802

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:           22,            22,             1;
  3:          320,           480,           162,             1;
  4:         6532,         13064,          7674,          1142,             1;
  5:       171752,        429380,        359504,        109876,          8002,            1;
  6:      5535256,      16605768,      18049008,       8421736,       1499262,        56022,           1;
  7:    211370240,     739795840,     988786320,     622476200,     181709920,     19986600,      392162,          1;
  8:   9333502096,   37334008384,   59221536976,   46995581584,   19255629388,   3741632584,   263299234,    2745142,        1;
  9: 467943326240, 2105744968080, 3866152258560, 3704723053920, 1962334509912, 557433565500, 74899480944, 3447204156, 19216002, 1;

%F Formula

The row polynomials satisfy P(n,x) = (6+6*x)*P(n-2,x) + (6*n-4+(6*n-5)*x)*P(n-1,x) - (6*x+6*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(6 + 6*x)*P[n - 2, x] + (6*n - 4 + (6*n - 5)*x)*P[n - 1, x] - (6*x + 6*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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (6 + 6*x)*v[i - 2] + (6*n - 4 + (6*n - 5)*x)*v[i - 1] - (6*x + 6*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) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

A144443 → x ↦ x/(x+1)

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