A082985 → D+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

\(\left(9\,x - 4\,x^{2} + 4\,n - 26\,n x + 8\,n x^{2} - 12\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(4 + 20\,x + 11\,x^{2} - 12\,x^{3} - 8\,n - 44\,n x - 46\,n x^{2} + 24\,n x^{3} - 4\,n^{2} - 4\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x + 9\,x^{2} - 16\,x^{4} - 2\,n x - 16\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	6	15	5													26
3	8	35	35	7												85
4	10	63	117	66	9											265
5	12	99	275	297	110	11										804
6	14	143	533	884	637	169	13									2393
7	16	195	915	2075	2340	1218	245	15								7019
8	18	255	1445	4182	6545	5406	2142	340	17							20350
9	20	323	2147	7581	15314	17765	11286	3534	456	19						58445
10	22	399	3045	12712	31605	47775	43043	21780	5544	595	21					166541
11	24	483	4163	20079	59432	111412	131859	95381	39468	8349	759	23				471432
12	26	575	5525	30250	104025	233580	345100	330200	196625	67925	12155	950	25			1326961
13	28	675	7155	43857	171990	450927	802332	963900	763776	381810	111969	17199	1170	27		3716815
14	30	783	9077	61596	271469	815045	1700937	2472888	2472888	1653522	704990	177944	23751	1421	29	10366370

Computed in your browser ?

Technical details

Node id: n-a8f047ca6c0b04d7 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d672f84bcb723328

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 0)
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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1:  4,   3;
  2:  6,  15,    5;
  3:  8,  35,   35,    7;
  4: 10,  63,  117,   66,     9;
  5: 12,  99,  275,  297,   110,    11;
  6: 14, 143,  533,  884,   637,   169,    13;
  7: 16, 195,  915, 2075,  2340,  1218,   245,   15;
  8: 18, 255, 1445, 4182,  6545,  5406,  2142,  340,  17;
  9: 20, 323, 2147, 7581, 15314, 17765, 11286, 3534, 456, 19;

%F Formula

The row polynomials satisfy (8*n^3-12*n^2+4*n+(16*n^2-26*n+9)*x+(8*n-4)*x^2)*P(n,x) = (8*n^3-4*n^2-8*n+4+(24*n^3-4*n^2-44*n+20)*x+(48*n^2-46*n+11)*x^2+(24*n-12)*x^3)*P(n-1,x) - ((4*n^2+2*n-2)*x+(16*n^2+16*n-9)*x^2+32*n*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 4*n^2 - 8*n + 4 + (24*n^3 - 4*n^2 - 44*n + 20)*x + (48*n^2 - 46*n + 11)*x^2 + (24*n - 12)*x^3)*P[n - 1, x] - ((4*n^2 + 2*n - 2)*x + (16*n^2 + 16*n - 9)*x^2 + 32*n*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 12*n^2 + 4*n + (16*n^2 - 26*n + 9)*x + (8*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^3 - 4*n^2 - 8*n + 4 + (24*n^3 - 4*n^2 - 44*n + 20)*x + (48*n^2 - 46*n + 11)*x^2 + (24*n - 12)*x^3)*v[i - 1] - ((4*n^2 + 2*n - 2)*x + (16*n^2 + 16*n - 9)*x^2 + 32*n*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 12*n^2 + 4*n + (16*n^2 - 26*n + 9)*x + (8*n - 4)*x^2)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-96ae67f48970d85a
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-96ae67f48970d85a
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-96ae67f48970d85a
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-96ae67f48970d85a

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image derivative_polynomial, shift 0 e-96ae67f48970d85a

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-4fa62233b528a616 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023