A114687 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

\(\left(-n - n^{2} + 2\,n^{2} x + 4\,n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(-n - 4\,n x + 4\,n x^{2} - n^{2} - 10\,n^{2} x + 16\,n^{2} x^{2} + 4\,n^{3} x + 32\,n^{3} x^{2} + 2\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-x + 4\,x^{2} - 4\,x^{3} - 2\,n x + 12\,n x^{2} - 16\,n x^{3} + 12\,n^{2} x^{2} - 24\,n^{2} x^{3} + 4\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	8							9
2	1	24	36						61
3	1	48	216	128					393
4	1	80	720	1280	400				2481
5	1	120	1800	6400	6000	1152			15473
6	1	168	3780	22400	42000	24192	3136		95677
7	1	224	7056	62720	196000	225792	87808	8192	587793

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-25c9e6499efaa423 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e9eb093223c2e083

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 ?

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: 1;
  1: 1,   8;
  2: 1,  24,    36;
  3: 1,  48,   216,    128;
  4: 1,  80,   720,   1280,     400;
  5: 1, 120,  1800,   6400,    6000,    1152;
  6: 1, 168,  3780,  22400,   42000,   24192,    3136;
  7: 1, 224,  7056,  62720,  196000,  225792,   87808,    8192;
  8: 1, 288, 12096, 150528,  705600, 1354752, 1053696,  294912,  20736;
  9: 1, 360, 19440, 322560, 2116800, 6096384, 7902720, 4423680, 933120, 51200;

%F Formula

The row polynomials satisfy -(n^2+n-(2*n^4+4*n^3+2*n^2)*x)*P(n,x) = -(n^2+n-(2*n^4+4*n^3-10*n^2-4*n)*x-(12*n^4+32*n^3+16*n^2+4*n)*x^2)*P(n-1,x) - ((2*n+1)*x-(4*n^3+12*n^2+12*n+4)*x^2+(8*n^3+24*n^2+16*n+4)*x^3)*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^2 + n - (2*n^4 + 4*n^3 - 10*n^2 - 4*n)*x - (12*n^4 + 32*n^3 + 16*n^2 + 4*n)*x^2)*P[n - 1, x] - ((2*n + 1)*x - (4*n^3 + 12*n^2 + 12*n + 4)*x^2 + (8*n^3 + 24*n^2 + 16*n + 4)*x^3)*D[P[n - 1, x], x])/(-n^2 - n + (2*n^4 + 4*n^3 + 2*n^2)*x)]];
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^2 + n - (2*n^4 + 4*n^3 - 10*n^2 - 4*n)*x - (12*n^4 + 32*n^3 + 16*n^2 + 4*n)*x^2)*v[i - 1] - ((2*n + 1)*x - (4*n^3 + 12*n^2 + 12*n + 4)*x^2 + (8*n^3 + 24*n^2 + 16*n + 4)*x^3)*deriv(v[i - 1], x))/(-n^2 - n + (2*n^4 + 4*n^3 + 2*n^2)*x)); 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-bd5326a0f3962370
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-bd5326a0f3962370
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-bd5326a0f3962370
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-bd5326a0f3962370

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 euler_polynomial, shift 0 e-bd5326a0f3962370

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-0f37534727a0e0cd root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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