A249250 → 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

\(\left(-2 - 8\,x - 14\,x^{2} - 3\,n - 10\,n x + 2\,n x^{2} - n x^{3} - n^{2} - 9\,n^{2} x + 9\,n^{2} x^{2} + n^{2} x^{3} - 3\,n^{3} x + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2 - 8\,x - 14\,x^{2} - 3\,n - 34\,n x - 85\,n x^{2} + 2\,n x^{3} - n^{2} - 21\,n^{2} x - 66\,n^{2} x^{2} - 2\,n^{2} x^{3} - 3\,n^{3} x - 15\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 35\,x^{2} + 153\,x^{3} + 25\,x^{4} + x^{5} + n x + 23\,n x^{2} + 132\,n x^{3} + 23\,n x^{4} + n x^{5} + 3\,n^{2} x^{2} + 30\,n^{2} x^{3} + 3\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	6							7
2	1	12	9						22
3	1	18	45	12					76
4	1	24	108	96	15				244
5	1	30	198	360	165	18			772
6	1	36	315	912	855	252	21		2392
7	1	42	459	1860	2895	1674	357	24	7312

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f6acc7b818ac622f · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c3e1b34dd6afb3e0

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,  6;
  2: 1, 12,   9;
  3: 1, 18,  45,   12;
  4: 1, 24, 108,   96,    15;
  5: 1, 30, 198,  360,   165,    18;
  6: 1, 36, 315,  912,   855,   252,    21;
  7: 1, 42, 459, 1860,  2895,  1674,   357,   24;
  8: 1, 48, 630, 3312,  7500,  7200,  2898,  480,  27;
  9: 1, 54, 828, 5376, 16290, 23148, 15204, 4608, 621, 30;

%F Formula

The row polynomials satisfy -(n^2+3*n+2+(3*n^3+9*n^2+10*n+8)*x-(3*n^3+9*n^2+2*n-14)*x^2-(n^2-n)*x^3)*P(n,x) = -(n^2+3*n+2+(3*n^3+21*n^2+34*n+8)*x+(15*n^3+66*n^2+85*n+14)*x^2+(2*n^2-2*n)*x^3)*P(n-1,x) + ((n+2)*x+(3*n^2+23*n+35)*x^2+(30*n^2+132*n+153)*x^3+(3*n^2+23*n+25)*x^4+(n+1)*x^5)*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 + 3*n + 2 + (3*n^3 + 21*n^2 + 34*n + 8)*x + (15*n^3 + 66*n^2 + 85*n + 14)*x^2 + (2*n^2 - 2*n)*x^3)*P[n - 1, x] + ((n + 2)*x + (3*n^2 + 23*n + 35)*x^2 + (30*n^2 + 132*n + 153)*x^3 + (3*n^2 + 23*n + 25)*x^4 + (n + 1)*x^5)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 - (3*n^3 + 9*n^2 + 10*n + 8)*x + (3*n^3 + 9*n^2 + 2*n - 14)*x^2 + (n^2 - n)*x^3)]];
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 + 3*n + 2 + (3*n^3 + 21*n^2 + 34*n + 8)*x + (15*n^3 + 66*n^2 + 85*n + 14)*x^2 + (2*n^2 - 2*n)*x^3)*v[i - 1] + ((n + 2)*x + (3*n^2 + 23*n + 35)*x^2 + (30*n^2 + 132*n + 153)*x^3 + (3*n^2 + 23*n + 25)*x^4 + (n + 1)*x^5)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 - (3*n^3 + 9*n^2 + 10*n + 8)*x + (3*n^3 + 9*n^2 + 2*n - 14)*x^2 + (n^2 - n)*x^3)); 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-195b29f840e130c6
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-195b29f840e130c6
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-195b29f840e130c6
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-195b29f840e130c6

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-195b29f840e130c6

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-86238c6a50aeb955 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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.