A055356 → (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

\(\left(-2 + n\right)\,P_{n}(x) = \left(-8\,x - 36\,x^{2} + 12\,n x + 72\,n x^{2} - 4\,n^{2} x - 45\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 8\,x^{2} - 16\,x^{3} - 2\,n x - 12\,n x^{2} + 32\,n x^{3} + 4\,n^{2} x^{2} - 20\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(x^{2} + 8\,x^{3} - n x^{2} - 12\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-2 + 8\,x + n - 12\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 8\,x^{2} + 2\,n x + 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	1	4						5
3	1	16	18					35
4	1	44	162	96				303
5	1	104	882	1536	600			3123
6	1	228	3816	13984	15000	4320		37349
7	1	480	14526	96640	206100	155520	35280	508547

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-14892a0af7e11cf4 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-8507039130aabebd

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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: 1,    4;
  3: 1,   16,     18;
  4: 1,   44,    162,      96;
  5: 1,  104,    882,    1536,      600;
  6: 1,  228,   3816,   13984,    15000,     4320;
  7: 1,  480,  14526,   96640,   206100,   155520,    35280;
  8: 1,  988,  51138,  567328,  2087500,  3010608,  1728720,   322560;
  9: 1, 2008, 171198, 2996608, 17529100, 42111648, 44883216, 20643840, 3265920;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -((4*n^2-12*n+8)*x-(9*n^3-45*n^2+72*n-36)*x^2)*P(n-2,x) - ((2*n-2)*x-(4*n^2-12*n+8)*x^2-(4*n^3-20*n^2+32*n-16)*x^3)*d/dx P(n-2,x) - ((n-1)*x^2-(4*n^2-12*n+8)*x^3)*d^2/dx^2 P(n-2,x) + (n-2+(4*n^2-12*n+8)*x)*P(n-1,x) + ((2*n-3)*x-(4*n^2-12*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 - 12*n + 8)*x - (9*n^3 - 45*n^2 + 72*n - 36)*x^2)*P[n - 2, x] - ((2*n - 2)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n^3 - 20*n^2 + 32*n - 16)*x^3)*D[P[n - 2, x], x] - ((n - 1)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], {x, 2}] + (n - 2 + (4*n^2 - 12*n + 8)*x)*P[n - 1, x] + ((2*n - 3)*x - (4*n^2 - 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
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; v[3] = 1 + 4*x; for(n = 3, N, my(i = n + 1); v[i] = (-((4*n^2 - 12*n + 8)*x - (9*n^3 - 45*n^2 + 72*n - 36)*x^2)*v[i - 2] - ((2*n - 2)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n^3 - 20*n^2 + 32*n - 16)*x^3)*deriv(v[i - 2], x) - ((n - 1)*x^2 - (4*n^2 - 12*n + 8)*x^3)*deriv(deriv(v[i - 2], x), x) + (n - 2 + (4*n^2 - 12*n + 8)*x)*v[i - 1] + ((2*n - 3)*x - (4*n^2 - 12*n + 8)*x^2)*deriv(v[i - 1], x))/(n - 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-00470390dfbbae49
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-00470390dfbbae49
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-00470390dfbbae49
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-00470390dfbbae49

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-00470390dfbbae49

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-b2d04b9d5c5e725d root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

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.