A290595 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-5 - 13\,x - 11\,x^{2} - 3\,x^{3} + 3\,n + 9\,n x + 9\,n x^{2} + 3\,n x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	3														4
3	4	27	27													58
4	28	306	675	405												1414
5	280	4314	15390	19845	8505											48334
6	3640	73224	370062	759780	688905	229635										2125246
7	58240	1456776	9679068	27875988	39803400	27785835	7577955									114237262
8	1106560	33272112	276939324	1041862500	2070185040	2258230590	1280674395	295540245								7257810766
9	24344320	858360240	8652648096	40605917220	104767753860	158081193270	139297968810	66496555125	13299311025							532084051966
10	608608000	24691383360	293982898176	1666138572648	5338219912740	10409190131340	12629222776170	9327250132200	3843500886225	678264862275						44211070163134
11	17041024000	783691124160	10810234188864	72230511522312	278508200842872	671800411982700	1049357803614120	1063237949733960	675551802825900	244853615281275	38661097149675					4105812359289838
12	528271744000	27207852013440	428191545457728	3310819594258176	15009534120180600	43435687989133440	83664952152261480	108805621370486040	94570198441058340	52707962447390250	17049543843006675	2435649120429525				421445896747419694
13	17961239296000	1025708307281280	18188768046833280	160349287462326144	839425451269925520	2848806349236207360	6565797599363402400	10502216591598852120	11686991927699088000	8889397615754305650	4411772440138012950	1288458384707218725	168059789309637225			47380507874132386654
14	664565853952000	41728820623795200	825196774919644800	8194561785374500224	48828612910418005392	190948628175048984240	514954410823742329440	983379025476447359760	1343039503085575495200	1305201938853318105600	882288996129031040550	394604385299028204300	105037368318523265625	12604484198222791875		5789949505216127474206
15	26582634158080000	1822251052689177600	39836848574374444800	441142972679493461376	2957010522395835791808	13138919646790266424368	40751427633754847209200	90883385203487828711760	147907406802493015357680	176089137106820295758400	151819163509911122124900	92330927848119293833500	37578168889634883510000	9188668980504415276875	1020963220056046141875	764148008018908565304142

Technical details

Node id: n-76d87447f0c0edac · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-6c53b1c3395f48c0

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         1;
   2:         1,           3;
   3:         4,          27,           27;
   4:        28,         306,          675,           405;
   5:       280,        4314,        15390,         19845,          8505;
   6:      3640,       73224,       370062,        759780,        688905,         229635;
   7:     58240,     1456776,      9679068,      27875988,      39803400,       27785835,        7577955;
   8:   1106560,    33272112,    276939324,    1041862500,    2070185040,     2258230590,     1280674395,     295540245;
   9:  24344320,   858360240,   8652648096,   40605917220,  104767753860,   158081193270,   139297968810,   66496555125,   13299311025;
  10: 608608000, 24691383360, 293982898176, 1666138572648, 5338219912740, 10409190131340, 12629222776170, 9327250132200, 3843500886225, 678264862275;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3*n-5+(9*n-13)*x+(9*n-11)*x^2+(3*n-3)*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 5 + (9*n - 13)*x + (9*n - 11)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n - 5 + (9*n - 13)*x + (9*n - 11)*x^2 + (3*n - 3)*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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