A257627 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 12\,x + 15\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(7\,x + 21\,x^{2} + 21\,x^{3} + 7\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	9	78	78													165
3	27	834	2340	1560												4761
4	81	8502	50622	84240	42120											185565
5	243	85506	971100	3034200	3580200	1432080										9103329
6	729	856518	17620278	92242800	192909600	176145840	58715280									538491045
7	2187	8569554	310679460	2566232760	8470753200	13252468320	9864167040	2818333440								37291205961
8	6561	85708662	5392955022	67803175440	331881633720	791599404960	987238717920	620033356800	155008339200							2959043298285
9	19683	857125986	92794447980	1735135311000	12119016382200	41364752934960	76881317909760	79629193013760	43247326636800	9610517030400						264680910812529
10	59049	8571377958	1588648253478	43499136423600	422538161245200	1983907168523280	5155614176272560	7831021296384000	6931197887328000	3315628375488000	663125675097600					26348129096452725
11	177147	85714133874	27118448222580	1075752239235960	14273159682038400	89754769886221440	313342831841719680	654682587873050880	839646846653472000	648565741796544000	277186532190796800	50397551307417600				2888952977633030361
12	531441	857142401622	462127903524222	26360422706114640	471513260602561320	3895968684865720320	17780372998212464640	49083950497801190400	85546706285572646400	94863443451087168000	64985653033889702400	25097980551093964800	4182996758515660800			345935408929393651005
13	1594323	8571427204866	7867317211132860	641892703017235800	15328642491744496200	164078260885384457040	959851500999055441920	3405823329799860203520	7746872948227851609600	11564883057477257164800	11289530269598962867200	6946748074663134566400	2447053103731661568000	376469708266409472000		44917288664732979014529
14	4782969	85714281614598	133855821142921878	15562771216636316400	492519020225544748800	6756147758363922238320	49920526241258257625040	223175685197547391795200	644385787084435556160000	1243588404583625772672000	1622112640646655033676800	1413744146462571943219200	789473710221694725427200	255622931912892031488000	36517561701841718784000	6285805757543863963470405

Technical details

Node id: n-4254826c474a0431 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-db62b20fb03c4b11

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:     3,         6;
  2:     9,        78,          78;
  3:    27,       834,        2340,          1560;
  4:    81,      8502,       50622,         84240,          42120;
  5:   243,     85506,      971100,       3034200,        3580200,        1432080;
  6:   729,    856518,    17620278,      92242800,      192909600,      176145840,       58715280;
  7:  2187,   8569554,   310679460,    2566232760,     8470753200,    13252468320,     9864167040,     2818333440;
  8:  6561,  85708662,  5392955022,   67803175440,   331881633720,   791599404960,   987238717920,   620033356800,   155008339200;
  9: 19683, 857125986, 92794447980, 1735135311000, 12119016382200, 41364752934960, 76881317909760, 79629193013760, 43247326636800, 9610517030400;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+12*x+15*x^2+6*x^3)*P(n-1,x) + (7*x+21*x^2+21*x^3+7*x^4)*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[Together[((3 + 12*x + 15*x^2 + 6*x^3)*P[n - 1, x] + (7*x + 21*x^2 + 21*x^3 + 7*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; for(n = 1, N, my(i = n + 1); v[i] = ((3 + 12*x + 15*x^2 + 6*x^3)*v[i - 1] + (7*x + 21*x^2 + 21*x^3 + 7*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) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257627 → 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