A008459 → 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

\(\left(2\,n - 4\,n x + n x^{2} - 2\,n^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(6\,x^{2} - 2\,x^{3} + n - 2\,n x - 12\,n x^{2} + 3\,n x^{3} - 4\,n^{2} x + 6\,n^{2} x^{2} + n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(-4\,x^{2} + 6\,x^{3} - 2\,x^{4} - n x + 5\,n x^{2} - 4\,n x^{3} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	5	6	1						12
3	10	27	12	1					50
4	17	88	84	20	1				210
5	26	225	400	200	30	1			882
6	37	486	1425	1300	405	42	1		3696
7	50	931	4116	6125	3430	735	56	1	15444

Show all 15 stored rows

Generated b-files: A092443 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-6457b4a2d865b72d · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a07eaa75d981d953

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 ?

A092443: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:  2,    1;
  2:  5,    6,     1;
  3: 10,   27,    12,     1;
  4: 17,   88,    84,    20,     1;
  5: 26,  225,   400,   200,    30,     1;
  6: 37,  486,  1425,  1300,   405,    42,     1;
  7: 50,  931,  4116,  6125,  3430,   735,    56,    1;
  8: 65, 1632, 10192, 22736, 20580,  7840,  1232,   72,  1;
  9: 82, 2673, 22464, 70560, 95256, 58212, 16128, 1944, 90, 1;

%F Formula

The row polynomials satisfy (n^3-2*n^2+2*n+(2*n^2-4*n)*x+n*x^2)*P(n,x) = (n^3+n+(3*n^3-4*n^2-2*n)*x+(6*n^2-12*n+6)*x^2+(3*n-2)*x^3)*P(n-1,x) + ((2*n^2-n)*x-(2*n^2-5*n+4)*x^2-(4*n-6)*x^3-2*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[((n^3 + n + (3*n^3 - 4*n^2 - 2*n)*x + (6*n^2 - 12*n + 6)*x^2 + (3*n - 2)*x^3)*P[n - 1, x] + ((2*n^2 - n)*x - (2*n^2 - 5*n + 4)*x^2 - (4*n - 6)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + 2*n + (2*n^2 - 4*n)*x + n*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] = ((n^3 + n + (3*n^3 - 4*n^2 - 2*n)*x + (6*n^2 - 12*n + 6)*x^2 + (3*n - 2)*x^3)*v[i - 1] + ((2*n^2 - n)*x - (2*n^2 - 5*n + 4)*x^2 - (4*n - 6)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + 2*n + (2*n^2 - 4*n)*x + n*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-cd6bcf4594ae6a7e
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-cd6bcf4594ae6a7e
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-cd6bcf4594ae6a7e
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-cd6bcf4594ae6a7e

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-cd6bcf4594ae6a7e

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Square the entries of Pascal's triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-c6b04c4560617585 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-1906182487ade7a7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square the entries of Pascal's triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-c6b04c4560617585 (expanded above)

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