A039756 → 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

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-1 + 6\,x - 3\,x^{2} - 2\,n + 8\,n x + 2\,n x^{2} - 8\,n^{2} x + 12\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x^{2} - 2\,x^{3} + 4\,n x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x + 4\,n - 8\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	8	3						12
3	1	18	39	4					62
4	1	32	174	160	5				372
5	1	50	510	1320	605	6			2492
6	1	72	1185	6080	8855	2184	7		18384
7	1	98	2373	20300	62055	55314	7651	8	147800

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-d28f63ae20e4d8dc · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1ea903b913797f87

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,   2;
  2: 1,   8,    3;
  3: 1,  18,   39,      4;
  4: 1,  32,  174,    160,       5;
  5: 1,  50,  510,   1320,     605,       6;
  6: 1,  72, 1185,   6080,    8855,    2184,       7;
  7: 1,  98, 2373,  20300,   62055,   55314,    7651,       8;
  8: 1, 128, 4284,  55104,  290430,  576576,  330316,   26240,     9;
  9: 1, 162, 7164, 129360, 1048110, 3713220, 5039020, 1913760, 88569, 10;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = -(2*n+1+(8*n^2-8*n-6)*x+(8*n^3-12*n^2-2*n+3)*x^2)*P(n-2,x) + ((4*n+2)*x^2+(8*n^2-2)*x^3)*d/dx P(n-2,x) + (4*n+(8*n^2-8*n+2)*x)*P(n-1,x) - (4*n-2)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 1 + (8*n^2 - 8*n - 6)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*P[n - 2, x] + ((4*n + 2)*x^2 + (8*n^2 - 2)*x^3)*D[P[n - 2, x], x] + (4*n + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] - (4*n - 2)*x^2*D[P[n - 1, x], x])/(2*n - 1)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 1 + (8*n^2 - 8*n - 6)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*v[i - 2] + ((4*n + 2)*x^2 + (8*n^2 - 2)*x^3)*deriv(v[i - 2], x) + (4*n + (8*n^2 - 8*n + 2)*x)*v[i - 1] - (4*n - 2)*x^2*deriv(v[i - 1], x))/(2*n - 1)); 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-f30f22ad14f353b2
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-f30f22ad14f353b2
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-f30f22ad14f353b2
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-f30f22ad14f353b2

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-f30f22ad14f353b2

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of B-analogs of Stirling numbers of 2nd kind.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-7519e7535554cb98 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.