A367270 → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2\,x^{2} + 5\,n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(6 + 4\,x - 16\,n - 8\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	3	2						5
3	10	12	3					25
4	35	60	30	4				129
5	126	280	210	60	5			681
6	462	1260	1260	560	105	6		3653
7	1716	5544	6930	4200	1260	168	7	19825

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

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

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

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

Technical details

Node id: n-c2b0d68b9e568dde · revision 2 · JSON
Definition source: derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-f0f102dbcc021c62)
Fingerprint: f6-2c1fea4a558d4e6c

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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 ?

A002002: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:     1;
  2:     3,      2;
  3:    10,     12,      3;
  4:    35,     60,     30,      4;
  5:   126,    280,    210,     60,     5;
  6:   462,   1260,   1260,    560,   105,     6;
  7:  1716,   5544,   6930,   4200,  1260,   168,    7;
  8:  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;
  9: 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;

%F Formula

The row polynomials satisfy (2*n^2-3*n)*P(n,x) = -(2*n^2-5*n+2)*x^2*P(n-2,x) + (8*n^2-16*n+6+(4*n^2-8*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 5*n + 2)*x^2*P[n - 2, x] + (8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x])/(2*n^2 - 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 5*n + 2)*x^2*v[i - 2] + (8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*v[i - 1])/(2*n^2 - 3*n)); 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-9efc939e395fae07
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-5e9a418c539b3915, e-9efc939e395fae07
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-5e9a418c539b3915, e-9efc939e395fae07
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-5e9a418c539b3915, e-9efc939e395fae07
Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-5e9a418c539b3915

Proof chains

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

basis_map A007318 e-9efc939e395fae07

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-2b43df6c63e34146 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
definition_equality e-f0f102dbcc021c62 root: theorem check

Ore-algebra identity of two definitions polytool ?
preserver_image affine, shift 0 e-5e9a418c539b3915

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - ↵ theorem_check three_term_recurrence e-2b43df6c63e34146 (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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023