A249252 → 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

\(\left(-2 - 12\,x - 32\,x^{2} - 3\,n - 10\,n x + 4\,n x^{2} - 8\,n x^{3} - n^{2} - 10\,n^{2} x + 20\,n^{2} x^{2} + 8\,n^{2} x^{3} - 4\,n^{3} x + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2 - 12\,x - 32\,x^{2} - 3\,n - 42\,n x - 132\,n x^{2} + 8\,n x^{3} - n^{2} - 26\,n^{2} x - 100\,n^{2} x^{2} - 8\,n^{2} x^{3} - 4\,n^{3} x - 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 42\,x^{2} + 228\,x^{3} + 120\,x^{4} + 16\,x^{5} + n x + 28\,n x^{2} + 200\,n x^{3} + 112\,n x^{4} + 16\,n x^{5} + 4\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	8							9
2	1	16	24						41
3	1	24	96	64					185
4	1	32	216	384	160				793
5	1	40	384	1216	1280	384			3305
6	1	48	600	2816	5280	3840	896		13481
7	1	56	864	5440	15360	19584	10752	2048	54105

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f1454451ff7417fc · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4b18ed72a763d464

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,  8;
  2: 1, 16,   24;
  3: 1, 24,   96,    64;
  4: 1, 32,  216,   384,   160;
  5: 1, 40,  384,  1216,  1280,    384;
  6: 1, 48,  600,  2816,  5280,   3840,    896;
  7: 1, 56,  864,  5440, 15360,  19584,  10752,   2048;
  8: 1, 64, 1176,  9344, 36000,  69120,  65408,  28672,  4608;
  9: 1, 72, 1536, 14784, 72960, 192384, 272384, 202752, 73728, 10240;

%F Formula

The row polynomials satisfy -(n^2+3*n+2+(4*n^3+10*n^2+10*n+12)*x-(8*n^3+20*n^2+4*n-32)*x^2-(8*n^2-8*n)*x^3)*P(n,x) = -(n^2+3*n+2+(4*n^3+26*n^2+42*n+12)*x+(24*n^3+100*n^2+132*n+32)*x^2+(8*n^2-8*n)*x^3)*P(n-1,x) + ((n+2)*x+(4*n^2+28*n+42)*x^2+(48*n^2+200*n+228)*x^3+(16*n^2+112*n+120)*x^4+(16*n+16)*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 + (4*n^3 + 26*n^2 + 42*n + 12)*x + (24*n^3 + 100*n^2 + 132*n + 32)*x^2 + (8*n^2 - 8*n)*x^3)*P[n - 1, x] + ((n + 2)*x + (4*n^2 + 28*n + 42)*x^2 + (48*n^2 + 200*n + 228)*x^3 + (16*n^2 + 112*n + 120)*x^4 + (16*n + 16)*x^5)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 - (4*n^3 + 10*n^2 + 10*n + 12)*x + (8*n^3 + 20*n^2 + 4*n - 32)*x^2 + (8*n^2 - 8*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 + (4*n^3 + 26*n^2 + 42*n + 12)*x + (24*n^3 + 100*n^2 + 132*n + 32)*x^2 + (8*n^2 - 8*n)*x^3)*v[i - 1] + ((n + 2)*x + (4*n^2 + 28*n + 42)*x^2 + (48*n^2 + 200*n + 228)*x^3 + (16*n^2 + 112*n + 120)*x^4 + (16*n + 16)*x^5)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 - (4*n^3 + 10*n^2 + 10*n + 12)*x + (8*n^3 + 20*n^2 + 4*n - 32)*x^2 + (8*n^2 - 8*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-422f06a6bd66498d
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-422f06a6bd66498d
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-422f06a6bd66498d
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-422f06a6bd66498d

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-422f06a6bd66498d

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-fb471901eb718e65 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.