A123519 → 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

\(\left(9 - 3\,x^{2} + 15\,n - 2\,n x^{2} + 9\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(21 + 20\,x - 7\,x^{2} - 4\,x^{3} + 29\,n + 30\,n x - 4\,n x^{2} - 2\,n x^{3} + 13\,n^{2} + 14\,n^{2} x + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x - 2\,x^{2} - 8\,x^{3} - 2\,x^{4} + 16\,n x + 8\,n x^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 7 + 14\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	7	14	4															25
1	13	52	44	8														117
2	21	140	228	120	16													525
3	31	310	812	800	304	32												2289
4	43	602	2296	3552	2480	736	64											9773
5	57	1064	5544	12240	13200	7104	1728	128										41065
6	73	1752	11928	35376	53680	44096	19264	3968	256									170393
7	91	2730	23496	89760	180752	206752	136640	50176	8960	512								699869
8	111	4070	43164	205920	528528	794976	725760	400384	126720	19968	1024							2850625
9	133	5852	74932	435864	1384240	2632448	3137792	2376192	1123584	312320	44032	2048						11529437
10	157	8164	124124	863720	3317600	7750912	11606784	11410944	7368960	3046400	754688	96256	4096					46352805
11	183	11102	197652	1619904	7390240	20749248	37953792	46636032	38916864	21880320	8031232	1794048	208896	8192				185397705
12	211	14770	304304	2899520	15480608	51328576	112248960	167691264	174155520	126038528	62695424	20684800	4206592	450560	16384			738216021
13	241	19280	455056	4985760	30770272	118781312	305325440	543012096	682978560	612964352	391198720	174387200	52236288	9748480	966656	32768		2927862481
14	273	24752	663408	8279136	58450080	259598976	773587584	1610844928	2403346176	2603673600	2054440960	1171722240	473001984	129712128	22364160	2064384	65536	11571840305

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Technical details

Node id: n-cc96bf3355920697 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6ba98ca01708d88c

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:   7,   14,     4;
  1:  13,   52,    44,      8;
  2:  21,  140,   228,    120,      16;
  3:  31,  310,   812,    800,     304,      32;
  4:  43,  602,  2296,   3552,    2480,     736,      64;
  5:  57, 1064,  5544,  12240,   13200,    7104,    1728,     128;
  6:  73, 1752, 11928,  35376,   53680,   44096,   19264,    3968,     256;
  7:  91, 2730, 23496,  89760,  180752,  206752,  136640,   50176,    8960,    512;
  8: 111, 4070, 43164, 205920,  528528,  794976,  725760,  400384,  126720,  19968,  1024;
  9: 133, 5852, 74932, 435864, 1384240, 2632448, 3137792, 2376192, 1123584, 312320, 44032, 2048;

%F Formula

The row polynomials satisfy (2*n^3+9*n^2+15*n+9-(2*n+3)*x^2)*P(n,x) = (2*n^3+13*n^2+29*n+21+(2*n^3+14*n^2+30*n+20)*x-(4*n+7)*x^2-(2*n+4)*x^3)*P(n-1,x) + ((4*n^2+16*n+12)*x+(2*n^2+8*n-2)*x^2-8*x^3-2*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 7 + 14*x + 4*x^2.

%t Mathematica

P[0, x_] := 7 + 14*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 13*n^2 + 29*n + 21 + (2*n^3 + 14*n^2 + 30*n + 20)*x - (4*n + 7)*x^2 - (2*n + 4)*x^3)*P[n - 1, x] + ((4*n^2 + 16*n + 12)*x + (2*n^2 + 8*n - 2)*x^2 - 8*x^3 - 2*x^4)*D[P[n - 1, x], x])/(2*n^3 + 9*n^2 + 15*n + 9 - (2*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 7 + 14*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + 13*n^2 + 29*n + 21 + (2*n^3 + 14*n^2 + 30*n + 20)*x - (4*n + 7)*x^2 - (2*n + 4)*x^3)*v[i - 1] + ((4*n^2 + 16*n + 12)*x + (2*n^2 + 8*n - 2)*x^2 - 8*x^3 - 2*x^4)*deriv(v[i - 1], x))/(2*n^3 + 9*n^2 + 15*n + 9 - (2*n + 3)*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-3fa839d676fcf5c7
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-3fa839d676fcf5c7
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-3fa839d676fcf5c7
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-3fa839d676fcf5c7

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-3fa839d676fcf5c7

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8d057b494ddd47e8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-2a4594def0262eef 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