A062264 → 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

\(\left(-4\,n - n^{2} + 16\,n^{2} x + 8\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(-24\,x + 48\,x^{2} - 4\,n - 20\,n x + 100\,n x^{2} - n^{2} + 10\,n^{2} x + 73\,n^{2} x^{2} + 8\,n^{3} x + 26\,n^{3} x^{2} + n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x + 20\,x^{2} - 16\,x^{3} - 2\,n x + 30\,n x^{2} - 28\,n x^{3} + 15\,n^{2} x^{2} - 15\,n^{2} x^{3} + 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	10							11
2	1	24	45						70
3	1	42	189	140					372
4	1	64	504	896	350				1815
5	1	90	1080	3360	3150	756			8437
6	1	120	2025	9600	15750	9072	1470		38038
7	1	154	3465	23100	57750	58212	22638	2640	167960

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-140881c726fb036b · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3b55b4d46620c501

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,  10;
  2: 1,  24,   45;
  3: 1,  42,  189,   140;
  4: 1,  64,  504,   896,    350;
  5: 1,  90, 1080,  3360,   3150,    756;
  6: 1, 120, 2025,  9600,  15750,   9072,    1470;
  7: 1, 154, 3465, 23100,  57750,  58212,   22638,   2640;
  8: 1, 192, 5544, 49280, 173250, 266112,  181104,  50688,   4455;
  9: 1, 234, 8424, 96096, 450450, 972972, 1009008, 494208, 104247, 7150;

%F Formula

The row polynomials satisfy -(n^2+4*n-(n^4+8*n^3+16*n^2)*x)*P(n,x) = -(n^2+4*n-(n^4+8*n^3+10*n^2-20*n-24)*x-(3*n^4+26*n^3+73*n^2+100*n+48)*x^2)*P(n-1,x) - ((2*n+4)*x-(2*n^3+15*n^2+30*n+20)*x^2+(2*n^3+15*n^2+28*n+16)*x^3)*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 + 4*n - (n^4 + 8*n^3 + 10*n^2 - 20*n - 24)*x - (3*n^4 + 26*n^3 + 73*n^2 + 100*n + 48)*x^2)*P[n - 1, x] - ((2*n + 4)*x - (2*n^3 + 15*n^2 + 30*n + 20)*x^2 + (2*n^3 + 15*n^2 + 28*n + 16)*x^3)*D[P[n - 1, x], x])/(-n^2 - 4*n + (n^4 + 8*n^3 + 16*n^2)*x)]];
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 + 4*n - (n^4 + 8*n^3 + 10*n^2 - 20*n - 24)*x - (3*n^4 + 26*n^3 + 73*n^2 + 100*n + 48)*x^2)*v[i - 1] - ((2*n + 4)*x - (2*n^3 + 15*n^2 + 30*n + 20)*x^2 + (2*n^3 + 15*n^2 + 28*n + 16)*x^3)*deriv(v[i - 1], x))/(-n^2 - 4*n + (n^4 + 8*n^3 + 16*n^2)*x)); 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-53c50964658e5225
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-53c50964658e5225
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-53c50964658e5225
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-53c50964658e5225

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-53c50964658e5225

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Coefficient triangle of certain polynomials N(4; m,x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-c22f7d753efa18e6 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.