A144441 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)

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

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

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	16	16													33
3	1	86	252	168												507
4	1	436	2852	4832	2416											10537
5	1	2186	28632	92688	110400	44160										278067
6	1	10936	272552	1504416	3205168	2943552	981184									8917809
7	1	54686	2527332	22406200	76034960	120319968	89905984	25687424								336936555
8	1	273436	23117852	317687872	1613388992	3914186944	4919283776	3098189824	774547456							14660676153
9	1	1367186	209920032	4371448928	31974977200	111632692320	209879626880	218589786880	118968134400	26437363200						722065317027
10	1	6835936	1898577152	59021601536	606138120288	2927560435776	7725020206784	11831308001280	10515711936000	5038589957120	1007717991424					39712973663297
11	1	34179686	17133678732	787111740360	11148975942240	72534641020992	258106265434752	545244506690304	703671611581440	545312265922560	233364478009344	42429905092608				2412616929293019
12	1	170898436	154435530452	10411411064352	200795836097872	1726569667288448	8062553532554880	22565463917892608	39656723695728896	44190776712652800	30352026911012864	11734839037353984	1955806506225664			160456121834301257
13	1	854492186	1391081883432	136961370574768	3562505950241120	39920032413549504	239883105462070784	865347409091135488	1988849202584332032	2988383008496023040	2928678553942349824	1805967593919197184	636740292055748608	97960044931653632		11595430102153251603
14	1	4272460936	12525547497752	1795027062021856	62522323728525008	903290616777528640	6883027811663403968	31371192098456142848	91714958027818811648	178439932952674627584	233969765036118745088	204571212206437228544	114443131094808219648	37083659566339391488	5297665652334198784	904742164944038803793

Technical details

Node id: n-26a4554f5dce94b2 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-bb41b383faa3577c

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,      16,        16;
  3: 1,      86,       252,        168;
  4: 1,     436,      2852,       4832,        2416;
  5: 1,    2186,     28632,      92688,      110400,        44160;
  6: 1,   10936,    272552,    1504416,     3205168,      2943552,       981184;
  7: 1,   54686,   2527332,   22406200,    76034960,    120319968,     89905984,     25687424;
  8: 1,  273436,  23117852,  317687872,  1613388992,   3914186944,   4919283776,   3098189824,    774547456;
  9: 1, 1367186, 209920032, 4371448928, 31974977200, 111632692320, 209879626880, 218589786880, 118968134400, 26437363200;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4*x+12*x^2+12*x^3+4*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*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[((4*x + 12*x^2 + 12*x^3 + 4*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*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] = ((4*x + 12*x^2 + 12*x^3 + 4*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*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 T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n,…?
  - 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 T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n,…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

A144441 → 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