A081580 → 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

\(\left(-n + 2\,n x + 11\,n x^{2} - 6\,n^{2} x + 6\,n^{2} x^{2}\right)\,P_{n}(x) = \left(22\,x + 124\,x^{2} + 22\,x^{3} - n - 22\,n x - 49\,n x^{2} + 12\,n x^{3} - 6\,n^{2} x - 66\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(x + 33\,x^{2} + 243\,x^{3} + 11\,x^{4} + 6\,n x^{2} + 132\,n x^{3} + 6\,n x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	14	3						18
3	1	26	39	4					70
4	1	38	183	76	5				303
5	1	50	435	580	125	6			1197
6	1	62	795	2380	1325	186	7		4756
7	1	74	1263	6340	7925	2526	259	8	18396

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ad50c5d2e676b791 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-cf4abaa1406d6942

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, 14,    3;
  3: 1, 26,   39,     4;
  4: 1, 38,  183,    76,     5;
  5: 1, 50,  435,   580,   125,      6;
  6: 1, 62,  795,  2380,  1325,    186,     7;
  7: 1, 74, 1263,  6340,  7925,   2526,   259,    8;
  8: 1, 86, 1839, 13324, 30725,  19986,  4291,  344,   9;
  9: 1, 98, 2523, 24196, 87005, 104406, 42343, 6728, 441, 10;

%F Formula

The row polynomials satisfy -(n+(6*n^2-2*n)*x-(6*n^2+11*n)*x^2)*P(n,x) = -(n+(6*n^2+22*n-22)*x+(66*n^2+49*n-124)*x^2-(12*n+22)*x^3)*P(n-1,x) + (x+(6*n+33)*x^2+(132*n+243)*x^3+(6*n+11)*x^4)*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 + (6*n^2 + 22*n - 22)*x + (66*n^2 + 49*n - 124)*x^2 - (12*n + 22)*x^3)*P[n - 1, x] + (x + (6*n + 33)*x^2 + (132*n + 243)*x^3 + (6*n + 11)*x^4)*D[P[n - 1, x], x])/(-n - (6*n^2 - 2*n)*x + (6*n^2 + 11*n)*x^2)]];
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 + (6*n^2 + 22*n - 22)*x + (66*n^2 + 49*n - 124)*x^2 - (12*n + 22)*x^3)*v[i - 1] + (x + (6*n + 33)*x^2 + (132*n + 243)*x^3 + (6*n + 11)*x^4)*deriv(v[i - 1], x))/(-n - (6*n^2 - 2*n)*x + (6*n^2 + 11*n)*x^2)); 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-5726a07dd01f8b02
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-5726a07dd01f8b02
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-5726a07dd01f8b02
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-5726a07dd01f8b02

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-5726a07dd01f8b02

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Pascal-(1,5,1) array.
  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-307703197d4a988b root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-bf7624d7e60440c3
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Pascal-(1,5,1) array.
    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-307703197d4a988b (expanded above)

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.