A159854 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 11\,x + 4\,x^{2} + 7\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(9 + 27\,x + 27\,x^{2} + 9\,x^{3} + 7\,n x + 13\,n x^{2} + 6\,n x^{3} + n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 9\,x^{2}\)

Rows ?

n	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	9															9
1	9	16														25
2	9	32	25													66
3	9	48	75	36												168
4	9	64	150	144	49											416
5	9	80	250	360	245	64										1008
6	9	96	375	720	735	384	81									2400
7	9	112	525	1260	1715	1344	567	100								5632
8	9	128	700	2016	3430	3584	2268	800	121							13056
9	9	144	900	3024	6174	8064	6804	3600	1089	144						29952
10	9	160	1125	4320	10290	16128	17010	12000	5445	1440	169					68096
11	9	176	1375	5940	16170	29568	37422	33000	19965	7920	1859	196				153600
12	9	192	1650	7920	24255	50688	74844	79200	59895	31680	11154	2352	225			344064
13	9	208	1950	10296	35035	82368	138996	171600	155727	102960	48334	15288	2925	256		765952
14	9	224	2275	13104	49049	128128	243243	343200	363363	288288	169169	71344	20475	3584	289	1695744

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Technical details

Node id: n-14222607db8216c9 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-598bfcebae6f1754

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

A058396: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 0, 0, 9;
  1: 0, 0, 9,  16;
  2: 0, 0, 9,  32,  25;
  3: 0, 0, 9,  48,  75,   36;
  4: 0, 0, 9,  64, 150,  144,   49;
  5: 0, 0, 9,  80, 250,  360,  245,   64;
  6: 0, 0, 9,  96, 375,  720,  735,  384,   81;
  7: 0, 0, 9, 112, 525, 1260, 1715, 1344,  567,  100;
  8: 0, 0, 9, 128, 700, 2016, 3430, 3584, 2268,  800,  121;
  9: 0, 0, 9, 144, 900, 3024, 6174, 8064, 6804, 3600, 1089, 144;

%F Formula

The row polynomials satisfy (9+(7*n+11)*x+(n^2+4*n+4)*x^2)*P(n,x) = (9+(7*n+27)*x+(n^2+13*n+27)*x^2+(n^2+6*n+9)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 9*x^2.

%t Mathematica

P[0, x_] := 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9 + (7*n + 27)*x + (n^2 + 13*n + 27)*x^2 + (n^2 + 6*n + 9)*x^3)*P[n - 1, x])/(9 + (7*n + 11)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((9 + (7*n + 27)*x + (n^2 + 13*n + 27)*x^2 + (n^2 + 6*n + 9)*x^3)*v[i - 1])/(9 + (7*n + 11)*x + (n^2 + 4*n + 4)*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-608e43d41fd9eb09
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-608e43d41fd9eb09
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-608e43d41fd9eb09

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-608e43d41fd9eb09

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Riordan array (1-x,x/(1-x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-b62074ba45057e82 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.