A207606 → 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

\(\left(-60 + 30\,x - 6\,x^{2} - 6\,x^{3} - 52\,n + 36\,n x - 6\,n x^{2} - 4\,n x^{3} - 18\,n^{2} + 18\,n^{2} x - 2\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-132 + 6\,x + 20\,x^{2} - 14\,x^{3} - 4\,x^{4} - 94\,n + 21\,n x + 24\,n x^{2} - 9\,n x^{3} - 2\,n x^{4} - 24\,n^{2} + 14\,n^{2} x + 14\,n^{2} x^{2} - 2\,n^{3} + 3\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-48\,x + 16\,x^{2} - 21\,x^{3} - 15\,x^{4} - 2\,x^{5} - 28\,n x + 25\,n x^{2} + 8\,n x^{3} - 4\,n^{2} x + 7\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	11	11	2															24
1	21	38	17	2														78
2	36	102	81	23	2													244
3	57	232	283	140	29	2												743
4	85	469	808	604	215	35	2											2218
5	121	868	1998	2080	1105	306	41	2										6521
6	166	1500	4434	6094	4460	1826	413	47	2									18942
7	221	2454	9042	15796	15119	8456	2807	536	53	2								54486
8	287	3839	17226	37180	44863	32571	14672	4088	675	59	2							155462
9	365	5786	31031	80938	119795	108920	63308	23808	5709	830	65	2						440557
10	456	8450	53339	165165	293540	325448	235620	113772	36660	7710	1001	71	2					1241234
11	561	12012	88101	319176	669630	886788	779076	466488	192213	54120	10131	1188	77	2				3479563
12	681	16681	140608	588744	1437826	2237098	2337228	1691304	860937	308913	77176	13012	1391	83	2			9711684
13	817	22696	217804	1043120	2930902	5285572	6461938	5540172	3396525	1500796	476410	106912	16393	1610	89	2		27001758
14	970	30328	328644	1784252	5710640	11802420	16663570	16661062	12063546	6400900	2495350	709722	144508	20314	1845	95	2	74818168

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Technical details

Node id: n-6ad40d76ed68b588 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a734f1d8065db00a

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

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:  11,   11,     2;
  1:  21,   38,    17,     2;
  2:  36,  102,    81,    23,      2;
  3:  57,  232,   283,   140,     29,      2;
  4:  85,  469,   808,   604,    215,     35,     2;
  5: 121,  868,  1998,  2080,   1105,    306,    41,     2;
  6: 166, 1500,  4434,  6094,   4460,   1826,   413,    47,    2;
  7: 221, 2454,  9042, 15796,  15119,   8456,  2807,   536,   53,   2;
  8: 287, 3839, 17226, 37180,  44863,  32571, 14672,  4088,  675,  59,  2;
  9: 365, 5786, 31031, 80938, 119795, 108920, 63308, 23808, 5709, 830, 65, 2;

%F Formula

The row polynomials satisfy -(2*n^3+18*n^2+52*n+60-(4*n^3+18*n^2+36*n+30)*x+(6*n+6)*x^2+(4*n+6)*x^3)*P(n,x) = -(2*n^3+24*n^2+94*n+132-(3*n^3+14*n^2+21*n+6)*x-(2*n^3+14*n^2+24*n+20)*x^2+(9*n+14)*x^3+(2*n+4)*x^4)*P(n-1,x) - ((4*n^2+28*n+48)*x-(7*n^2+25*n+16)*x^2-(2*n^2+8*n-21)*x^3+15*x^4+2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 11*x + 2*x^2.

%t Mathematica

P[0, x_] := 11 + 11*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 24*n^2 + 94*n + 132 - (3*n^3 + 14*n^2 + 21*n + 6)*x - (2*n^3 + 14*n^2 + 24*n + 20)*x^2 + (9*n + 14)*x^3 + (2*n + 4)*x^4)*P[n - 1, x] - ((4*n^2 + 28*n + 48)*x - (7*n^2 + 25*n + 16)*x^2 - (2*n^2 + 8*n - 21)*x^3 + 15*x^4 + 2*x^5)*D[P[n - 1, x], x])/(-2*n^3 - 18*n^2 - 52*n - 60 + (4*n^3 + 18*n^2 + 36*n + 30)*x - (6*n + 6)*x^2 - (4*n + 6)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 11 + 11*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 + 24*n^2 + 94*n + 132 - (3*n^3 + 14*n^2 + 21*n + 6)*x - (2*n^3 + 14*n^2 + 24*n + 20)*x^2 + (9*n + 14)*x^3 + (2*n + 4)*x^4)*v[i - 1] - ((4*n^2 + 28*n + 48)*x - (7*n^2 + 25*n + 16)*x^2 - (2*n^2 + 8*n - 21)*x^3 + 15*x^4 + 2*x^5)*deriv(v[i - 1], x))/(-2*n^3 - 18*n^2 - 52*n - 60 + (4*n^3 + 18*n^2 + 36*n + 30)*x - (6*n + 6)*x^2 - (4*n + 6)*x^3)); 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-da807e70e77c2903
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-da807e70e77c2903
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-da807e70e77c2903
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-da807e70e77c2903

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-da807e70e77c2903

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-62ac8b7225a515c7 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3222de37fcca30b3 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