A059344 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n x + n^{2} x\right)\,P_{n}(x) = \left(1 - 4\,x - 4\,x^{2} + 5\,n x - 2\,n x^{2} + n^{2} x + 4\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2} - 8\,x^{3} - 12\,n x^{3} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	4							5
3	1	12							13
4	1	24	36						61
5	1	40	180						221
6	1	60	540	480					1081
7	1	84	1260	3360					4705
8	1	112	2520	13440	8400				24473
9	1	144	4536	40320	75600				120601
10	1	180	7560	100800	378000	181440			667981
11	1	220	11880	221760	1386000	1995840			3615701
12	1	264	17820	443520	4158000	11975040	4656960		21251605
13	1	312	25740	823680	10810800	51891840	60540480		124092853
14	1	364	36036	1441440	25225200	181621440	423783360	138378240	770486081

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Technical details

Node id: n-fd271958fce1b7c5 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-eb658cc66b4f4727

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2: 1,   4;
  3: 1,  12;
  4: 1,  24,   36;
  5: 1,  40,  180;
  6: 1,  60,  540,   480;
  7: 1,  84, 1260,  3360;
  8: 1, 112, 2520, 13440,  8400;
  9: 1, 144, 4536, 40320, 75600;

%F Formula

The row polynomials satisfy (1+(n^2+n)*x)*P(n,x) = (1+(n^2+5*n-4)*x+(2*n^3+4*n^2-2*n-4)*x^2)*P(n-1,x) - (4*x^2+(4*n^2+12*n+8)*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[((1 + (n^2 + 5*n - 4)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - (4*x^2 + (4*n^2 + 12*n + 8)*x^3)*D[P[n - 1, x], x])/(1 + (n^2 + n)*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] = ((1 + (n^2 + 5*n - 4)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*v[i - 1] - (4*x^2 + (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 1], x))/(1 + (n^2 + n)*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-0f564e87cb0e9c13
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-0f564e87cb0e9c13
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-0f564e87cb0e9c13

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-0f564e87cb0e9c13

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers paper_theorem e-94e95c2ea0369f05 root: theorem check
    
    Parent-row provers of polytool polytool ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.