A135838 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + x^{2} + 3\,n x - 2\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 12\,x^{2} + 8\,x^{3} + 2\,x^{4} + 6\,n x + 16\,n x^{2} + 14\,n x^{3} + 4\,n x^{4} + 2\,n^{2} x^{2} + 4\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	8							10
2	2	16	18						36
3	4	48	108	64					224
4	4	64	216	256	100				640
5	8	160	720	1280	1000	288			3456
6	8	192	1080	2560	3000	1728	392		8960
7	16	448	3024	8960	14000	12096	5488	1024	45056

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2c954f8eb27a6499 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d42e6339eab21b9c

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:  2,    8;
  2:  2,   16,    18;
  3:  4,   48,   108,    64;
  4:  4,   64,   216,   256,    100;
  5:  8,  160,   720,  1280,   1000,    288;
  6:  8,  192,  1080,  2560,   3000,   1728,    392;
  7: 16,  448,  3024,  8960,  14000,  12096,   5488,  1024;
  8: 16,  512,  4032, 14336,  28000,  32256,  21952,  8192,  1296;
  9: 32, 1152, 10368, 43008, 100800, 145152, 131712, 73728, 23328, 3200;

%F Formula

The row polynomials satisfy (1+(3*n-4)*x+(n^2-2*n+1)*x^2)*P(n,x) = (2+(6*n+8)*x+(2*n^2+16*n+12)*x^2+(4*n^2+14*n+8)*x^3+(2*n^2+4*n+2)*x^4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n + 8)*x + (2*n^2 + 16*n + 12)*x^2 + (4*n^2 + 14*n + 8)*x^3 + (2*n^2 + 4*n + 2)*x^4)*P[n - 2, x])/(1 + (3*n - 4)*x + (n^2 - 2*n + 1)*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; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((2 + (6*n + 8)*x + (2*n^2 + 16*n + 12)*x^2 + (4*n^2 + 14*n + 8)*x^3 + (2*n^2 + 4*n + 2)*x^4)*v[i - 2])/(1 + (3*n - 4)*x + (n^2 - 2*n + 1)*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-064e625d7496a635
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-064e625d7496a635
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-064e625d7496a635

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-064e625d7496a635

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image reversal, shift 0 e-07f7d31230240b98
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-51cb02553cf4a8ac root: theorem check
    
    Parent-row provers of polytool polytool ?
  - ↵ parent_provers periodic_product e-51cb02553cf4a8ac (expanded above)

Related sequences ?

Same fingerprint (12)

Equal up to content, sign, reversal and a small index shift.

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.