A116414 → 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

\(\left(-16\,n - 8\,n x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-32 - 28\,x - 8\,x^{2} - 32\,n - 32\,n x - 8\,n x^{2} + n x^{3} - 4\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3} + 4\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-16 - 48\,x - 36\,x^{2} - 4\,x^{3} + 4\,n x^{2} + 8\,n x^{3} + n x^{4} + 4\,n^{2} x^{2} + 8\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	13	16	3						32
3	40	84	36	4					164
4	121	368	261	64	5				819
5	364	1462	1488	592	100	6			4012
6	1093	5472	7362	4160	1125	144	7		19363
7	3280	19688	33192	24680	9400	1908	196	8	92352

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-4d7f7ce12c5849de · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-38dc41046e8afe97

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:     4,      2;
  2:    13,     16,      3;
  3:    40,     84,     36,      4;
  4:   121,    368,    261,     64,      5;
  5:   364,   1462,   1488,    592,    100,      6;
  6:  1093,   5472,   7362,   4160,   1125,    144,     7;
  7:  3280,  19688,  33192,  24680,   9400,   1908,   196,    8;
  8:  9841,  68896, 140214, 130496,  65075,  18480,  2989,  256,   9;
  9: 29524, 236202, 564624, 634896, 395220, 146286, 32928, 4416, 324, 10;

%F Formula

The row polynomials satisfy -(16*n+8*n*x-2*n^3*x^2)*P(n,x) = -(32*n+32+(4*n^2+32*n+28)*x-(4*n^3+4*n^2-8*n-8)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (16+48*x-(4*n^2+4*n-36)*x^2-(8*n^2+8*n-4)*x^3-(n^2+n)*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[(-(32*n + 32 + (4*n^2 + 32*n + 28)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (16 + 48*x - (4*n^2 + 4*n - 36)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-16*n - 8*n*x + 2*n^3*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] = (-(32*n + 32 + (4*n^2 + 32*n + 28)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (16 + 48*x - (4*n^2 + 4*n - 36)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-16*n - 8*n*x + 2*n^3*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-4a2e695cedf93066
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-4a2e695cedf93066
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-4a2e695cedf93066
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-4a2e695cedf93066

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-4a2e695cedf93066

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-894010de7c322dc8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-7bedda0241d23786 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