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	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

Show all 15 stored rows

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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