A091044 → 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

\(\left(-n + n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(-1 + 2\,x - x^{2} - n - 5\,n x - 2\,n x^{2} + 3\,n^{2} x + 9\,n^{2} x^{2} + 2\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x + 4\,x^{2} - 2\,x^{3} + 8\,n x^{2} - 8\,n x^{3} + 4\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	3	20	9						32
3	4	56	84	16					160
4	5	120	378	240	25				768
5	6	220	1188	1584	550	36			3584
6	7	364	3003	6864	5005	1092	49		16384
7	8	560	6552	22880	28600	13104	1960	64	73728

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7b9205ef354b4cd5 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6d973dca5f5aeef0

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 ?

A292044: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  2,    4;
  2:  3,   20,     9;
  3:  4,   56,    84,     16;
  4:  5,  120,   378,    240,     25;
  5:  6,  220,  1188,   1584,    550,     36;
  6:  7,  364,  3003,   6864,   5005,   1092,     49;
  7:  8,  560,  6552,  22880,  28600,  13104,   1960,    64;
  8:  9,  816, 12852,  63648, 121550,  95472,  29988,  3264,   81;
  9: 10, 1140, 23256, 155040, 419900, 503880, 271320, 62016, 5130, 100;

%F Formula

The row polynomials satisfy -(n-(2*n^3+n^2)*x)*P(n,x) = -(n+1-(2*n^3+3*n^2-5*n+2)*x-(6*n^3+9*n^2-2*n-1)*x^2)*P(n-1,x) - (2*x-(4*n^2+8*n+4)*x^2+(4*n^2+8*n+2)*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 + 1 - (2*n^3 + 3*n^2 - 5*n + 2)*x - (6*n^3 + 9*n^2 - 2*n - 1)*x^2)*P[n - 1, x] - (2*x - (4*n^2 + 8*n + 4)*x^2 + (4*n^2 + 8*n + 2)*x^3)*D[P[n - 1, x], x])/(-n + (2*n^3 + 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 + 1 - (2*n^3 + 3*n^2 - 5*n + 2)*x - (6*n^3 + 9*n^2 - 2*n - 1)*x^2)*v[i - 1] - (2*x - (4*n^2 + 8*n + 4)*x^2 + (4*n^2 + 8*n + 2)*x^3)*deriv(v[i - 1], x))/(-n + (2*n^3 + 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-6b1fb9b9f3839b5c
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-6b1fb9b9f3839b5c
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-6b1fb9b9f3839b5c
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-6b1fb9b9f3839b5c

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-6b1fb9b9f3839b5c

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-75f779345b39f910
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023