A134991 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 13\,x^{2} + 12\,x^{3} + 4\,x^{4} - 2\,n x^{2} - 4\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)

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

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	4														5
3	1	12	26													39
4	1	28	158	236												423
5	1	60	664	2412	2752											5889
6	1	124	2404	15928	42460	39208										100125
7	1	252	8070	86760	404940	850896	660032									2010951
8	1	508	25962	425220	3088020	11063856	19167488	12818912								46589967
9	1	1020	81428	1954404	20708340	112389696	325857560	479980608	282137824							1223110881
10	1	2044	251408	8617616	127971816	986127576	4250740424	10334373008	13235325376	6939897856						35883307125
11	1	4092	768514	36951840	748368376	7860903120	47211888752	168302252784	351972811040	398650597056	188666182784					1163450728359
12	1	8188	2334166	155418844	4209919576	58621986976	470859960032	2306470543616	6997616508944	12831931162976	13025605275200	5617349020544				41312822139063
13	1	16380	7059792	644853276	23028027864	416603933568	4350898598704	28107219625920	115931712570576	305800809531264	499085825025792	458951042951424	181790703209728			1594458495404289
14	1	32764	21294012	2649650664	123382036068	2856464688264	38003316482896	314745071108896	1693855120598736	6023023851029568	14042315470188288	20641525709106432	17348124430825984	6353726042486272		66458301529528845
15	1	65532	64111358	10810756440	650887170708	19059907693392	318075167808064	3308330096482656	22581276432350480	103881241033753536	324240658348000416	676952843687442432	905008176016546048	700311038991421440	238513970965257728	2975135332408860231

Technical details

Node id: n-a93e05281575f401 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-11f5029b9713167f

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,    4;
   3: 1,   12,     26;
   4: 1,   28,    158,     236;
   5: 1,   60,    664,    2412,      2752;
   6: 1,  124,   2404,   15928,     42460,     39208;
   7: 1,  252,   8070,   86760,    404940,    850896,     660032;
   8: 1,  508,  25962,  425220,   3088020,  11063856,   19167488,    12818912;
   9: 1, 1020,  81428, 1954404,  20708340, 112389696,  325857560,   479980608,   282137824;
  10: 1, 2044, 251408, 8617616, 127971816, 986127576, 4250740424, 10334373008, 13235325376, 6939897856;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+6*x-(2*n-13)*x^2-(4*n-12)*x^3-(2*n-4)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*x^5)*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[((1 + 6*x - (2*n - 13)*x^2 - (4*n - 12)*x^3 - (2*n - 4)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*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] = ((1 + 6*x - (2*n - 13)*x^2 - (4*n - 12)*x^3 - (2*n - 4)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*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 of Ward numbers T(n,k) read by rows.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

Related ?

Same fingerprint ?

A269939 → x ↦ x/(x+1)

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741