A078812 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-2 - 12\,x - 10\,x^{2} + 2\,n + 9\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)

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

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	1	6	8													15
4	1	9	25	21												56
5	1	12	51	90	55											209
6	1	15	86	234	300	144										780
7	1	18	130	480	951	954	377									2911
8	1	21	183	855	2305	3573	2939	987								10864
9	1	24	245	1386	4740	10008	12707	8850	2584							40545
10	1	27	316	2100	8715	23373	40426	43398	26195	6765						151316
11	1	30	396	3024	14770	48132	105805	154668	143682	76500	17711					564719
12	1	33	485	4185	23526	90342	241486	448710	567260	464148	221016	46368				2107560
13	1	36	583	5610	35685	157896	497742	1125036	1807585	2011260	1469778	632916	121393			7865521
14	1	39	690	7326	52030	260766	947904	2527920	4941207	6985305	6936298	4578102	1799125	317811		29354524
15	1	42	806	9360	73425	411246	1694517	5213736	12027225	20683890	26084628	23375736	14063653	5082270	832040	109552575

Technical details

Node id: n-e9f75ddaed25275f · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-05007905755e7549

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: 1,  6,   8;
   4: 1,  9,  25,   21;
   5: 1, 12,  51,   90,   55;
   6: 1, 15,  86,  234,  300,   144;
   7: 1, 18, 130,  480,  951,   954,   377;
   8: 1, 21, 183,  855, 2305,  3573,  2939,   987;
   9: 1, 24, 245, 1386, 4740, 10008, 12707,  8850,  2584;
  10: 1, 27, 316, 2100, 8715, 23373, 40426, 43398, 26195, 6765;

%F Formula

The row polynomials satisfy (2*n-2)*P(n,x) = (2*n-2+(9*n-12)*x+(5*n-10)*x^2)*P(n-1,x) - (x+6*x^2+5*x^3)*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[((2*n - 2 + (9*n - 12)*x + (5*n - 10)*x^2)*P[n - 1, x] - (x + 6*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 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] = ((2*n - 2 + (9*n - 12)*x + (5*n - 10)*x^2)*v[i - 1] - (x + 6*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n - 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 basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    ↑ Three-term recurrence criterion (above)

Related ?

Candidate equalities ?

A078812 → reverse → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

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