A207606 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(18 + 24\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(18 + 69\,x + 57\,x^{2} + 24\,n + 100\,n x + 95\,n x^{2} + 15\,n x^{3} + 8\,n^{2} + 36\,n^{2} x + 38\,n^{2} x^{2} + 15\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-6\,x - 45\,x^{2} - 57\,x^{3} - 4\,n x - 30\,n x^{2} - 56\,n x^{3} - 30\,n x^{4}\right)\,P_{n-1}'(x) + \left(3\,x^{3} + 18\,x^{4} + 15\,x^{5}\right)\,P_{n-1}''(x)\)

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

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-4a8b92b99ec82596 · revision 0 · JSON
Definition source: derived:reversal
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)*P(n,x) = (8*n^2+24*n+18+(36*n^2+100*n+69)*x+(38*n^2+95*n+57)*x^2+(15*n^2+15*n)*x^3)*P(n-1,x) - ((4*n+6)*x+(30*n+45)*x^2+(56*n+57)*x^3+30*n*x^4)*d/dx P(n-1,x) + (3*x^3+18*x^4+15*x^5)*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 + (36*n^2 + 100*n + 69)*x + (38*n^2 + 95*n + 57)*x^2 + (15*n^2 + 15*n)*x^3)*P[n - 1, x] - ((4*n + 6)*x + (30*n + 45)*x^2 + (56*n + 57)*x^3 + 30*n*x^4)*D[P[n - 1, x], x] + (3*x^3 + 18*x^4 + 15*x^5)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18)]];
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 + (36*n^2 + 100*n + 69)*x + (38*n^2 + 95*n + 57)*x^2 + (15*n^2 + 15*n)*x^3)*v[i - 1] - ((4*n + 6)*x + (30*n + 45)*x^2 + (56*n + 57)*x^3 + 30*n*x^4)*deriv(v[i - 1], x) + (3*x^3 + 18*x^4 + 15*x^5)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18)); 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 basis_map_A007318 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula sec…?
  - 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
  - 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)

Related ?

Candidate equalities ?

A207606 → reverse → x ↦ x/(x+1) 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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023