A111125 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,x - 4\,n + 13\,n x - 9\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(2 + 2\,x - 4\,n + 3\,n x + 2\,n x^{2} - 7\,n^{2} x - 3\,n^{2} x^{2} + 2\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 2\,x^{2} - 8\,n x^{2} - 2\,n x^{3} + 4\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	2							5
3	5	10	3						18
4	7	28	21	4					60
5	9	60	81	36	5				191
6	11	110	231	176	55	6			589
7	13	182	546	624	325	78	7		1775
8	15	280	1134	1800	1375	540	105	8	5257

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-cb54800b91762264 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-24a6b574ee77e562

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 ?

A377857: row sums, 15 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:  1;
   2:  3,   2;
   3:  5,  10,    3;
   4:  7,  28,   21,     4;
   5:  9,  60,   81,    36,     5;
   6: 11, 110,  231,   176,    55,     6;
   7: 13, 182,  546,   624,   325,    78,    7;
   8: 15, 280, 1134,  1800,  1375,   540,  105,    8;
   9: 17, 408, 2142,  4488,  4675,  2652,  833,  136,   9;
  10: 19, 570, 3762, 10032, 13585, 10374, 4655, 1216, 171, 10;

%F Formula

The row polynomials satisfy -(4*n-6-(2*n^3-9*n^2+13*n-6)*x)*P(n,x) = -(4*n-2-(2*n^3-7*n^2+3*n+2)*x-(n^3-3*n^2+2*n)*x^2)*P(n-1,x) - (8*x-(4*n^2-8*n-2)*x^2-(n^2-2*n)*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 - (2*n^3 - 7*n^2 + 3*n + 2)*x - (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 1, x] - (8*x - (4*n^2 - 8*n - 2)*x^2 - (n^2 - 2*n)*x^3)*D[P[n - 1, x], x])/(-4*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*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 - (2*n^3 - 7*n^2 + 3*n + 2)*x - (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 1] - (8*x - (4*n^2 - 8*n - 2)*x^2 - (n^2 - 2*n)*x^3)*deriv(v[i - 1], x))/(-4*n + 6 + (2*n^3 - 9*n^2 + 13*n - 6)*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-1de9fee97176e47c
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-1de9fee97176e47c
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-1de9fee97176e47c
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-1de9fee97176e47c

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-1de9fee97176e47c

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-0bb8ed02ac535b32 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-19ca7a5e6e7a3cfc 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