A207823 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

\(\left(1152\,n + 96\,n x + 32\,n x^{2} - 240\,n^{3} x^{2} - 40\,n^{3} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(2304 + 1488\,x + 304\,x^{2} - 8\,x^{3} - 8\,x^{4} + 2304\,n + 2496\,n x + 256\,n x^{2} - 212\,n x^{3} - 44\,n x^{4} + 1008\,n^{2} x - 528\,n^{2} x^{2} - 464\,n^{2} x^{3} - 72\,n^{2} x^{4} + n^{2} x^{5} - 480\,n^{3} x^{2} - 284\,n^{3} x^{3} - 36\,n^{3} x^{4} + 3\,n^{3} x^{5} - 24\,n^{4} x^{3} + 4\,n^{4} x^{4} + 3\,n^{4} x^{5} + 4\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(1728 + 1728\,x + 432\,x^{2} - 64\,x^{3} - 40\,x^{4} - 4\,x^{5} - 1008\,n x^{2} - 912\,n x^{3} - 244\,n x^{4} - 20\,n x^{5} - 1008\,n^{2} x^{2} - 912\,n^{2} x^{3} - 232\,n^{2} x^{4} - 12\,n^{2} x^{5} + n^{2} x^{6} + 24\,n^{3} x^{4} + 16\,n^{3} x^{5} + 2\,n^{3} x^{6} + 12\,n^{4} x^{4} + 8\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	4							8
2	15	32	9						56
3	56	184	108	16					364
4	209	928	837	256	25				2255
5	780	4364	5328	2496	500	36			13504
6	2911	19648	30294	19200	5875	864	49		78841
7	10864	85872	160056	128160	53000	11880	1372	64	451268

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-185b66629b765b00 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-80ceda54e61f4898

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:      4,       4;
  2:     15,      32,       9;
  3:     56,     184,     108,      16;
  4:    209,     928,     837,     256,      25;
  5:    780,    4364,    5328,    2496,     500,      36;
  6:   2911,   19648,   30294,   19200,    5875,     864,     49;
  7:  10864,   85872,  160056,  128160,   53000,   11880,   1372,    64;
  8:  40545,  367296,  803142,  777984,  406375,  122976,  21609,  2048,   81;
  9: 151316, 1545492, 3878928, 4411584, 2788100, 1065204, 252448, 36352, 2916, 100;

%F Formula

The row polynomials satisfy (1152*n+96*n*x-(240*n^3-32*n)*x^2-40*n^3*x^3+2*n^5*x^4)*P(n,x) = (2304*n+2304+(1008*n^2+2496*n+1488)*x-(480*n^3+528*n^2-256*n-304)*x^2-(24*n^4+284*n^3+464*n^2+212*n+8)*x^3+(4*n^5+4*n^4-36*n^3-72*n^2-44*n-8)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (1728+1728*x-(1008*n^2+1008*n-432)*x^2-(912*n^2+912*n+64)*x^3+(12*n^4+24*n^3-232*n^2-244*n-40)*x^4+(8*n^4+16*n^3-12*n^2-20*n-4)*x^5+(n^4+2*n^3+n^2)*x^6)*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[((2304*n + 2304 + (1008*n^2 + 2496*n + 1488)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (24*n^4 + 284*n^3 + 464*n^2 + 212*n + 8)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (1728 + 1728*x - (1008*n^2 + 1008*n - 432)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (12*n^4 + 24*n^3 - 232*n^2 - 244*n - 40)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(1152*n + 96*n*x - (240*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)]];
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] = ((2304*n + 2304 + (1008*n^2 + 2496*n + 1488)*x - (480*n^3 + 528*n^2 - 256*n - 304)*x^2 - (24*n^4 + 284*n^3 + 464*n^2 + 212*n + 8)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (1728 + 1728*x - (1008*n^2 + 1008*n - 432)*x^2 - (912*n^2 + 912*n + 64)*x^3 + (12*n^4 + 24*n^3 - 232*n^2 - 244*n - 40)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(1152*n + 96*n*x - (240*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*x^4)); 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-9b17678c30419d79
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-9b17678c30419d79
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-9b17678c30419d79
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-9b17678c30419d79

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-9b17678c30419d79

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-720cd5df5ac0e3b3 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image affine, shift 0 e-56c0c02612a8191c
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)
    
    theorem_check three_term_recurrence e-c7df247a48c8552e root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-ec629e395a3def0c root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - basis_map A007318 e-9c2f7b8109ebb5e1
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-af8f298ca8f41adf root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
    
    ↵ theorem_check three_term_recurrence e-c7df247a48c8552e (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