A144443 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6\,x + 18\,x^{2} + 18\,x^{3} + 6\,x^{4}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(6\,x + 18\,x^{2} + 18\,x^{3} + 6\,x^{4}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	22	22													45
3	1	162	480	320												963
4	1	1142	7674	13064	6532											28413
5	1	8002	109876	359504	429380	171752										1078515
6	1	56022	1499262	8421736	18049008	16605768	5535256									50167053
7	1	392162	19986600	181709920	622476200	988786320	739795840	211370240								2764517283
8	1	2745142	263299234	3741632584	19255629388	46995581584	59221536976	37334008384	9333502096							176147935389
9	1	19216002	3447204156	74899480944	557433565500	1962334509912	3704723053920	3866152258560	2105744968080	467943326240						12742697583315
10	1	134512022	44983852902	1472947262376	15457858347288	75463626082104	200266964578056	307661325878400	273871536084720	131304136410080	26260827282016					1031804340289965
11	1	941584162	585981479920	28636570446720	416375484040320	2745070723853376	9839832332467872	20873584012120320	27002228858675760	20951135690150240	8970384232933376	1630978951442432				92458813779194499
12	1	6591089142	7626098984394	552569289396104	10991226096634068	96024539858160960	452461588371195648	1273132124698913856	2244594480351145056	2505913103063829920	1722888090892318176	666384801970664832	111064133661777472			9084014290944109629
13	1	46137624002	99197665015236	10609103422691984	286007373515333620	3265202686442397768	19832832722442751872	72013731138496657920	166187223522025846560	250342199563292674880	245715027503298840256	151646372486896772352	53485456470421218880	8228531764680187520		971013293578738012851
14	1	322963368022	1289978292724942	203007528481802216	7365727578667463168	108726737763429185960	838943391292267318008	3853528999177557001344	11322131684951540086944	22099288511712689799872	29036282898746108898752	25420011468990348732032	14230727288806907396224	4612642481834699665024	658948925976385666432	112188802414660339108941

Technical details

Node id: n-a37a8a8ecf8890f6 · revision 0 · JSON
Definition source: derived:mobius_x_over_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: 1,        2;
  2: 1,       22,         22;
  3: 1,      162,        480,         320;
  4: 1,     1142,       7674,       13064,         6532;
  5: 1,     8002,     109876,      359504,       429380,        171752;
  6: 1,    56022,    1499262,     8421736,     18049008,      16605768,       5535256;
  7: 1,   392162,   19986600,   181709920,    622476200,     988786320,     739795840,     211370240;
  8: 1,  2745142,  263299234,  3741632584,  19255629388,   46995581584,   59221536976,   37334008384,    9333502096;
  9: 1, 19216002, 3447204156, 74899480944, 557433565500, 1962334509912, 3704723053920, 3866152258560, 2105744968080, 467943326240;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6*x+18*x^2+18*x^3+6*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (6*x+18*x^2+18*x^3+6*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*x + 18*x^2 + 18*x^3 + 6*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*x + 18*x^2 + 18*x^3 + 6*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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, 0]
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)

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+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