A114656 → (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

\(\left(2\,n - 2\,n^{2} + n^{2} x - 2\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(-8\,x - 2\,x^{2} + 4\,n + 20\,n x + 5\,n x^{2} - 4\,n^{2} - 10\,n^{2} x - 2\,n^{2} x^{2} - 4\,n^{3} x - 4\,n^{3} x^{2} + 2\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(4\,x - x^{3} - 8\,n x + 2\,n x^{3} + 4\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	4							6
3	4	24	9						37
4	8	96	108	16					228
5	16	320	720	320	25				1401
6	32	960	3600	3200	750	36			8578
7	64	2688	15120	22400	10500	1512	49		52333
8	128	7168	56448	125440	98000	28224	2744	64	318216

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-dcdf634434d2b81e · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5ab569ff3c1291af

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   2,     4;
   3:   4,    24,      9;
   4:   8,    96,    108,      16;
   5:  16,   320,    720,     320,      25;
   6:  32,   960,   3600,    3200,     750,      36;
   7:  64,  2688,  15120,   22400,   10500,    1512,     49;
   8: 128,  7168,  56448,  125440,   98000,   28224,   2744,     64;
   9: 256, 18432, 193536,  602112,  705600,  338688,  65856,   4608,   81;
  10: 512, 46080, 622080, 2580480, 4233600, 3048192, 987840, 138240, 7290, 100;

%F Formula

The row polynomials satisfy -(2*n^2-2*n-(n^4-2*n^3+n^2)*x)*P(n,x) = -(4*n^2-4*n-(2*n^4-4*n^3-10*n^2+20*n-8)*x-(3*n^4-4*n^3-2*n^2+5*n-2)*x^2)*P(n-1,x) - ((8*n-4)*x-4*n^3*x^2+(2*n^3-2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 4*n - (2*n^4 - 4*n^3 - 10*n^2 + 20*n - 8)*x - (3*n^4 - 4*n^3 - 2*n^2 + 5*n - 2)*x^2)*P[n - 1, x] - ((8*n - 4)*x - 4*n^3*x^2 + (2*n^3 - 2*n + 1)*x^3)*D[P[n - 1, x], x])/(-2*n^2 + 2*n + (n^4 - 2*n^3 + n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(4*n^2 - 4*n - (2*n^4 - 4*n^3 - 10*n^2 + 20*n - 8)*x - (3*n^4 - 4*n^3 - 2*n^2 + 5*n - 2)*x^2)*v[i - 1] - ((8*n - 4)*x - 4*n^3*x^2 + (2*n^3 - 2*n + 1)*x^3)*deriv(v[i - 1], x))/(-2*n^2 + 2*n + (n^4 - 2*n^3 + n^2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-e1d01b2b90edc2fe
Coefficients of one sign. Every nonzero row, n >= 1: 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-e1d01b2b90edc2fe
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-e1d01b2b90edc2fe
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-e1d01b2b90edc2fe

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-e1d01b2b90edc2fe

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 >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-e6c0603bd7c28948 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A114655 → (θ+1)²

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