A081577 → 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

\(\left(-n + 2\,n x + 5\,n x^{2} - 3\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n}(x) = \left(10\,x + 28\,x^{2} + 10\,x^{3} - n - 10\,n x - 7\,n x^{2} + 6\,n x^{3} - 3\,n^{2} x - 15\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(x + 15\,x^{2} + 51\,x^{3} + 5\,x^{4} + 3\,n x^{2} + 30\,n x^{3} + 3\,n x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	8	3						12
3	1	14	21	4					40
4	1	20	66	40	5				132
5	1	26	138	184	65	6			420
6	1	32	237	544	395	96	7		1312
7	1	38	363	1228	1535	726	133	8	4032

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-06d4d24415e9e4e9 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-23446e8e74871f81

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,  8,   3;
  3: 1, 14,  21,    4;
  4: 1, 20,  66,   40,     5;
  5: 1, 26, 138,  184,    65,     6;
  6: 1, 32, 237,  544,   395,    96,    7;
  7: 1, 38, 363, 1228,  1535,   726,  133,    8;
  8: 1, 44, 516, 2344,  4430,  3516, 1204,  176,   9;
  9: 1, 50, 696, 4000, 10430, 12516, 7000, 1856, 225, 10;

%F Formula

The row polynomials satisfy -(n+(3*n^2-2*n)*x-(3*n^2+5*n)*x^2)*P(n,x) = -(n+(3*n^2+10*n-10)*x+(15*n^2+7*n-28)*x^2-(6*n+10)*x^3)*P(n-1,x) + (x+(3*n+15)*x^2+(30*n+51)*x^3+(3*n+5)*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 + (3*n^2 + 10*n - 10)*x + (15*n^2 + 7*n - 28)*x^2 - (6*n + 10)*x^3)*P[n - 1, x] + (x + (3*n + 15)*x^2 + (30*n + 51)*x^3 + (3*n + 5)*x^4)*D[P[n - 1, x], x])/(-n - (3*n^2 - 2*n)*x + (3*n^2 + 5*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 + (3*n^2 + 10*n - 10)*x + (15*n^2 + 7*n - 28)*x^2 - (6*n + 10)*x^3)*v[i - 1] + (x + (3*n + 15)*x^2 + (30*n + 51)*x^3 + (3*n + 5)*x^4)*deriv(v[i - 1], x))/(-n - (3*n^2 - 2*n)*x + (3*n^2 + 5*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-f58bd164c0caee20
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-f58bd164c0caee20
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-f58bd164c0caee20
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-f58bd164c0caee20

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-f58bd164c0caee20

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
  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-1f8dc4b132ba6116 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-8efc9aafb30c2198
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
    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-1f8dc4b132ba6116 (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.