A062145 → 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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	8							9
3	1	20	30						51
4	1	36	135	80					252
5	1	56	378	560	175				1170
6	1	80	840	2240	1750	336			5247
7	1	108	1620	6720	9450	4536	588		23023
8	1	140	2835	16800	36750	31752	10290	960	99528

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-3036ce9cc391cfe8 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-fb006a6f3bd92843

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: 1,   8;
   3: 1,  20,   30;
   4: 1,  36,  135,    80;
   5: 1,  56,  378,   560,    175;
   6: 1,  80,  840,  2240,   1750,    336;
   7: 1, 108, 1620,  6720,   9450,   4536,    588;
   8: 1, 140, 2835, 16800,  36750,  31752,  10290,    960;
   9: 1, 176, 4620, 36960, 115500, 155232,  90552,  21120,  1485;
  10: 1, 216, 7128, 73920, 311850, 598752, 543312, 228096, 40095, 2200;

%F Formula

The row polynomials satisfy -(n^2+n-2-(n^4+2*n^3-3*n^2-4*n+4)*x)*P(n,x) = -(n^2+n-2-(n^4+2*n^3-9*n^2-6*n)*x-(3*n^4+8*n^3+9*n-2)*x^2)*P(n-1,x) - ((2*n+1)*x-(2*n^3+6*n^2+2*n+2)*x^2+(2*n^3+6*n^2+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[(-(n^2 + n - 2 - (n^4 + 2*n^3 - 9*n^2 - 6*n)*x - (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*P[n - 1, x] - ((2*n + 1)*x - (2*n^3 + 6*n^2 + 2*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*D[P[n - 1, x], x])/(-n^2 - n + 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*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] = (-(n^2 + n - 2 - (n^4 + 2*n^3 - 9*n^2 - 6*n)*x - (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*v[i - 1] - ((2*n + 1)*x - (2*n^3 + 6*n^2 + 2*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*deriv(v[i - 1], x))/(-n^2 - n + 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*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-78be3a4437894f86
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-78be3a4437894f86
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-78be3a4437894f86
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-78be3a4437894f86

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-78be3a4437894f86

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-62efc507a1cd208a 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.