A249252 → (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

\(\left(1 + 6\,x + 28\,x^{2} - 24\,x^{3} + 2\,n + 20\,n x + 112\,n x^{2} + 64\,n x^{3} + n^{2} + 28\,n^{2} x + 112\,n^{2} x^{2} + 64\,n^{2} x^{3} + 16\,n^{2} x^{4} + 12\,n^{3} x + 64\,n^{3} x^{2} + 16\,n^{3} x^{3} + 16\,n^{4} x^{2}\right)\,P_{n}(x) = \left(8\,x + 132\,x^{2} + 664\,x^{3} + 240\,x^{4} + 32\,x^{5} + 8\,n x + 224\,n x^{2} + 1184\,n x^{3} + 480\,n x^{4} + 64\,n x^{5} + 2\,n^{2} x + 128\,n^{2} x^{2} + 800\,n^{2} x^{3} + 224\,n^{2} x^{4} + 32\,n^{2} x^{5} + 24\,n^{3} x^{2} + 256\,n^{3} x^{3} + 32\,n^{3} x^{4} + 32\,n^{4} x^{3}\right)\,P_{n-2}(x) + \left(1 + 14\,x + 160\,x^{2} + 640\,x^{3} + 240\,x^{4} + 32\,x^{5} + 2\,n + 44\,n x + 304\,n x^{2} + 992\,n x^{3} + 480\,n x^{4} + 64\,n x^{5} + n^{2} + 42\,n^{2} x + 280\,n^{2} x^{2} + 624\,n^{2} x^{3} + 208\,n^{2} x^{4} + 32\,n^{2} x^{5} + 12\,n^{3} x + 120\,n^{3} x^{2} + 208\,n^{3} x^{3} + 32\,n^{3} x^{4} + 16\,n^{4} x^{2} + 32\,n^{4} x^{3}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 1 + 16\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	16							17
2	1	32	72						105
3	1	48	288	256					593
4	1	64	648	1536	800				3049
5	1	80	1152	4864	6400	2304			14801
6	1	96	1800	11264	26400	23040	6272		68873
7	1	112	2592	21760	76800	117504	75264	16384	310417

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-804b8a7d74ebdfd0 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-2f9781f166a9a3d3

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,  16;
  2: 1,  32,   72;
  3: 1,  48,  288,   256;
  4: 1,  64,  648,  1536,    800;
  5: 1,  80, 1152,  4864,   6400,    2304;
  6: 1,  96, 1800, 11264,  26400,   23040,    6272;
  7: 1, 112, 2592, 21760,  76800,  117504,   75264,   16384;
  8: 1, 128, 3528, 37376, 180000,  414720,  457856,  229376,  41472;
  9: 1, 144, 4608, 59136, 364800, 1154304, 1906688, 1622016, 663552, 102400;

%F Formula

The row polynomials satisfy (n^2+2*n+1+(12*n^3+28*n^2+20*n+6)*x+(16*n^4+64*n^3+112*n^2+112*n+28)*x^2+(16*n^3+64*n^2+64*n-24)*x^3+16*n^2*x^4)*P(n,x) = ((2*n^2+8*n+8)*x+(24*n^3+128*n^2+224*n+132)*x^2+(32*n^4+256*n^3+800*n^2+1184*n+664)*x^3+(32*n^3+224*n^2+480*n+240)*x^4+(32*n^2+64*n+32)*x^5)*P(n-2,x) + (n^2+2*n+1+(12*n^3+42*n^2+44*n+14)*x+(16*n^4+120*n^3+280*n^2+304*n+160)*x^2+(32*n^4+208*n^3+624*n^2+992*n+640)*x^3+(32*n^3+208*n^2+480*n+240)*x^4+(32*n^2+64*n+32)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n^2 + 8*n + 8)*x + (24*n^3 + 128*n^2 + 224*n + 132)*x^2 + (32*n^4 + 256*n^3 + 800*n^2 + 1184*n + 664)*x^3 + (32*n^3 + 224*n^2 + 480*n + 240)*x^4 + (32*n^2 + 64*n + 32)*x^5)*P[n - 2, x] + (n^2 + 2*n + 1 + (12*n^3 + 42*n^2 + 44*n + 14)*x + (16*n^4 + 120*n^3 + 280*n^2 + 304*n + 160)*x^2 + (32*n^4 + 208*n^3 + 624*n^2 + 992*n + 640)*x^3 + (32*n^3 + 208*n^2 + 480*n + 240)*x^4 + (32*n^2 + 64*n + 32)*x^5)*P[n - 1, x])/(n^2 + 2*n + 1 + (12*n^3 + 28*n^2 + 20*n + 6)*x + (16*n^4 + 64*n^3 + 112*n^2 + 112*n + 28)*x^2 + (16*n^3 + 64*n^2 + 64*n - 24)*x^3 + 16*n^2*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; v[2] = 1 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (((2*n^2 + 8*n + 8)*x + (24*n^3 + 128*n^2 + 224*n + 132)*x^2 + (32*n^4 + 256*n^3 + 800*n^2 + 1184*n + 664)*x^3 + (32*n^3 + 224*n^2 + 480*n + 240)*x^4 + (32*n^2 + 64*n + 32)*x^5)*v[i - 2] + (n^2 + 2*n + 1 + (12*n^3 + 42*n^2 + 44*n + 14)*x + (16*n^4 + 120*n^3 + 280*n^2 + 304*n + 160)*x^2 + (32*n^4 + 208*n^3 + 624*n^2 + 992*n + 640)*x^3 + (32*n^3 + 208*n^2 + 480*n + 240)*x^4 + (32*n^2 + 64*n + 32)*x^5)*v[i - 1])/(n^2 + 2*n + 1 + (12*n^3 + 28*n^2 + 20*n + 6)*x + (16*n^4 + 64*n^3 + 112*n^2 + 112*n + 28)*x^2 + (16*n^3 + 64*n^2 + 64*n - 24)*x^3 + 16*n^2*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-9dcceef23b62a48e
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-9dcceef23b62a48e
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-9dcceef23b62a48e
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-9dcceef23b62a48e

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-9dcceef23b62a48e

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  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-fb471901eb718e65 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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.