A207606 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(18 + 72\,x + 108\,x^{2} + 72\,x^{3} + 18\,x^{4} + 24\,n + 96\,n x + 144\,n x^{2} + 96\,n x^{3} + 24\,n x^{4} + 8\,n^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} + 32\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n}(x) = \left(18 + 141\,x + 441\,x^{2} + 714\,x^{3} + 636\,x^{4} + 297\,x^{5} + 57\,x^{6} + 24\,n + 196\,n x + 639\,n x^{2} + 1091\,n x^{3} + 1054\,n x^{4} + 570\,n x^{5} + 155\,n x^{6} + 15\,n x^{7} + 8\,n^{2} + 68\,n^{2} x + 230\,n^{2} x^{2} + 415\,n^{2} x^{3} + 440\,n^{2} x^{4} + 278\,n^{2} x^{5} + 98\,n^{2} x^{6} + 15\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(-6\,x - 69\,x^{2} - 273\,x^{3} - 522\,x^{4} - 528\,x^{5} - 273\,x^{6} - 57\,x^{7} - 4\,n x - 46\,n x^{2} - 200\,n x^{3} - 450\,n x^{4} - 580\,n x^{5} - 434\,n x^{6} - 176\,n x^{7} - 30\,n x^{8}\right)\,P_{n-1}'(x) + \left(3\,x^{3} + 30\,x^{4} + 105\,x^{5} + 180\,x^{6} + 165\,x^{7} + 78\,x^{8} + 15\,x^{9}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 2 + 11\,x + 13\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	2	11	13															26
1	2	17	44	34														97
2	2	23	93	155	89													362
3	2	29	160	417	510	233												1351
4	2	35	245	874	1668	1608	610											5042
5	2	41	348	1580	4130	6195	4924	1597										18817
6	2	47	469	2589	8625	17711	21841	14761	4181									70226
7	2	53	608	3955	16044	42006	70844	74089	43540	10946								262087
8	2	59	765	5732	27440	87549	188237	268910	243966	126805	28657							978122
9	2	65	940	7974	44028	165914	434840	791615	979852	784614	365532	75025						3650401
10	2	71	1133	10735	67185	292266	905142	2008586	3166460	3455260	2475408	1044816	196418					13623482
11	2	77	1344	14069	98450	485847	1737912	4558098	8757378	12163025	11861336	7686426	2965334	514229				50843527
12	2	83	1573	18030	139524	770462	3128268	9479568	21526530	36426573	45183951	39815174	23549204	8365415	1346269			189750626
13	2	89	1820	22672	192270	1174965	5341204	18378525	48227322	96448065	145706292	163204002	131133390	71326601	23477180	3524578		708158977
14	2	95	2085	28049	258713	1733745	8726575	33631675	100234629	231650463	413523957	563896305	575561445	424911597	213907779	65590703	9227465	2642885282

Technical details

Node id: n-91c62c75d8f18841 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-797bb38c8aa87a31

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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: 2, 11,  13;
  1: 2, 17,  44,   34;
  2: 2, 23,  93,  155,    89;
  3: 2, 29, 160,  417,   510,    233;
  4: 2, 35, 245,  874,  1668,   1608,    610;
  5: 2, 41, 348, 1580,  4130,   6195,   4924,   1597;
  6: 2, 47, 469, 2589,  8625,  17711,  21841,  14761,   4181;
  7: 2, 53, 608, 3955, 16044,  42006,  70844,  74089,  43540,  10946;
  8: 2, 59, 765, 5732, 27440,  87549, 188237, 268910, 243966, 126805,  28657;
  9: 2, 65, 940, 7974, 44028, 165914, 434840, 791615, 979852, 784614, 365532, 75025;

%F Formula

The row polynomials satisfy (8*n^2+24*n+18+(32*n^2+96*n+72)*x+(48*n^2+144*n+108)*x^2+(32*n^2+96*n+72)*x^3+(8*n^2+24*n+18)*x^4)*P(n,x) = (8*n^2+24*n+18+(68*n^2+196*n+141)*x+(230*n^2+639*n+441)*x^2+(415*n^2+1091*n+714)*x^3+(440*n^2+1054*n+636)*x^4+(278*n^2+570*n+297)*x^5+(98*n^2+155*n+57)*x^6+(15*n^2+15*n)*x^7)*P(n-1,x) - ((4*n+6)*x+(46*n+69)*x^2+(200*n+273)*x^3+(450*n+522)*x^4+(580*n+528)*x^5+(434*n+273)*x^6+(176*n+57)*x^7+30*n*x^8)*d/dx P(n-1,x) + (3*x^3+30*x^4+105*x^5+180*x^6+165*x^7+78*x^8+15*x^9)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2 + 11*x + 13*x^2.

%t Mathematica

P[0, x_] := 2 + 11*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 + 24*n + 18 + (68*n^2 + 196*n + 141)*x + (230*n^2 + 639*n + 441)*x^2 + (415*n^2 + 1091*n + 714)*x^3 + (440*n^2 + 1054*n + 636)*x^4 + (278*n^2 + 570*n + 297)*x^5 + (98*n^2 + 155*n + 57)*x^6 + (15*n^2 + 15*n)*x^7)*P[n - 1, x] - ((4*n + 6)*x + (46*n + 69)*x^2 + (200*n + 273)*x^3 + (450*n + 522)*x^4 + (580*n + 528)*x^5 + (434*n + 273)*x^6 + (176*n + 57)*x^7 + 30*n*x^8)*D[P[n - 1, x], x] + (3*x^3 + 30*x^4 + 105*x^5 + 180*x^6 + 165*x^7 + 78*x^8 + 15*x^9)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18 + (32*n^2 + 96*n + 72)*x + (48*n^2 + 144*n + 108)*x^2 + (32*n^2 + 96*n + 72)*x^3 + (8*n^2 + 24*n + 18)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 11*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^2 + 24*n + 18 + (68*n^2 + 196*n + 141)*x + (230*n^2 + 639*n + 441)*x^2 + (415*n^2 + 1091*n + 714)*x^3 + (440*n^2 + 1054*n + 636)*x^4 + (278*n^2 + 570*n + 297)*x^5 + (98*n^2 + 155*n + 57)*x^6 + (15*n^2 + 15*n)*x^7)*v[i - 1] - ((4*n + 6)*x + (46*n + 69)*x^2 + (200*n + 273)*x^3 + (450*n + 522)*x^4 + (580*n + 528)*x^5 + (434*n + 273)*x^6 + (176*n + 57)*x^7 + 30*n*x^8)*deriv(v[i - 1], x) + (3*x^3 + 30*x^4 + 105*x^5 + 180*x^6 + 165*x^7 + 78*x^8 + 15*x^9)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18 + (32*n^2 + 96*n + 72)*x + (48*n^2 + 144*n + 108)*x^2 + (32*n^2 + 96*n + 72)*x^3 + (8*n^2 + 24*n + 18)*x^4)); 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 reversal of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? 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
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

A207606 → x ↦ x+1 → reverse 15 rows; no all-index equality certificate was found

Same fingerprint ?

A207606 → x ↦ x+1

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023