A144439 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

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

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	10	10	1													21
3	64	96	34	1												195
4	532	1064	638	106	1											2341
5	5448	13620	11536	3684	322	1										34611
6	66440	199320	218272	104344	19922	970	1									609269
7	941056	3293696	4425552	2829640	857952	104136	2914	1								12454947
8	15189776	60759104	96759856	77622704	32578908	6672264	534278	8746	1							290125637
9	275298080	1238841360	2281463040	2203860960	1187014008	348512460	50125616	2712204	26242	1						7587853971
10	5536341152	27681705760	57878749680	65424764160	43114763800	16620781032	3552030152	368238584	13683482	78730	1					220191136533
11	122350101504	672925558272	1574810512480	2039705619120	1591759905536	763012354720	219146169024	35019855680	2665247872	68784816	236194	1				7021464345219
12	2947475118400	17684850710400	45798767884192	66882707908960	60360233069536	34678042638784	12589260262016	2768646953536	337264528228	19096968424	345026318	708586	1			244066691777381
13	76879053281408	499713846329152	1418819931932928	2306657316010432	2364518576571200	1585682188923168	698384600297472	197392478898048	33903131978280	3193624566820	135886978896	1728438324	2125762	1		9185282366331891
14	2158508442116224	15109559094813568	46671491761808256	83604682338273152	95984661049600064	73658208906886464	38225285422255008	13269979523465856	2979151632377512	405583420422584	29868600922032	962270952024	8652111842	6377290	1	372097951122381877

Technical details

Node id: n-16dc446fc8a01758 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-42d134b85075e1e7

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:        10,         10,          1;
  3:        64,         96,         34,          1;
  4:       532,       1064,        638,        106,          1;
  5:      5448,      13620,      11536,       3684,        322,         1;
  6:     66440,     199320,     218272,     104344,      19922,       970,        1;
  7:    941056,    3293696,    4425552,    2829640,     857952,    104136,     2914,       1;
  8:  15189776,   60759104,   96759856,   77622704,   32578908,   6672264,   534278,    8746,     1;
  9: 275298080, 1238841360, 2281463040, 2203860960, 1187014008, 348512460, 50125616, 2712204, 26242, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + (2*n+(2*n-1)*x)*P(n-1,x) - (2*x+2*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[(2 + 2*x)*P[n - 2, x] + (2*n + (2*n - 1)*x)*P[n - 1, x] - (2*x + 2*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] = (2 + 2*x)*v[i - 2] + (2*n + (2*n - 1)*x)*v[i - 1] - (2*x + 2*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 T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A144439 → 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