A207607 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 42\,x + 12\,n - 37\,n x + 8\,n x^{2} - 3\,n^{2} x - 12\,n^{2} x^{2} - 2\,n^{3} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(6 + 102\,x + 40\,x^{2} + 4\,x^{3} + 12\,n - 25\,n x - 24\,n x^{2} - 2\,n x^{3} - 3\,n^{2} x - 19\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{3} x + 3\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(24\,x - 82\,x^{2} - 38\,x^{3} - 4\,x^{4} - 8\,n x^{2} - 10\,n x^{3} - 2\,n x^{4} - 4\,n^{2} x^{2} + 7\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	10	6													17
3	1	18	27	8												54
4	1	28	75	52	10											166
5	1	40	165	196	85	12										499
6	1	54	315	560	405	126	14									1475
7	1	70	546	1344	1425	726	175	16								4303
8	1	88	882	2856	4125	3036	1183	232	18							12421
9	1	108	1350	5544	10395	10296	5733	1800	297	20						35544
10	1	130	1980	10032	23595	30030	22295	9920	2601	370	22					100976
11	1	154	2805	17160	49335	78078	73892	43520	16065	3610	451	24				285095
12	1	180	3861	28028	96525	185328	216580	161568	78489	24700	4851	540	26			800677
13	1	208	5187	44044	178750	408408	575484	527136	322677	133000	36421	6348	637	28		2238329
14	1	238	6825	66976	316030	845988	1410864	1550400	1159893	599830	214291	51888	8125	742	30	6232121

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Technical details

Node id: n-758be4c7caa6b7b4 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-ba328c78c6169b80

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,   4;
  2: 1,  10,    6;
  3: 1,  18,   27,    8;
  4: 1,  28,   75,   52,    10;
  5: 1,  40,  165,  196,    85,    12;
  6: 1,  54,  315,  560,   405,   126,   14;
  7: 1,  70,  546, 1344,  1425,   726,  175,   16;
  8: 1,  88,  882, 2856,  4125,  3036, 1183,  232,  18;
  9: 1, 108, 1350, 5544, 10395, 10296, 5733, 1800, 297, 20;

%F Formula

The row polynomials satisfy (12*n+6-(2*n^3+3*n^2+37*n-42)*x+(4*n^3-12*n^2+8*n)*x^2)*P(n,x) = (12*n+6-(2*n^3+3*n^2+25*n-102)*x+(3*n^3-19*n^2-24*n+40)*x^2+(2*n^3-4*n^2-2*n+4)*x^3)*P(n-1,x) + (24*x-(4*n^2+8*n+82)*x^2+(7*n^2-10*n-38)*x^3+(2*n^2-2*n-4)*x^4)*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[((12*n + 6 - (2*n^3 + 3*n^2 + 25*n - 102)*x + (3*n^3 - 19*n^2 - 24*n + 40)*x^2 + (2*n^3 - 4*n^2 - 2*n + 4)*x^3)*P[n - 1, x] + (24*x - (4*n^2 + 8*n + 82)*x^2 + (7*n^2 - 10*n - 38)*x^3 + (2*n^2 - 2*n - 4)*x^4)*D[P[n - 1, x], x])/(12*n + 6 - (2*n^3 + 3*n^2 + 37*n - 42)*x + (4*n^3 - 12*n^2 + 8*n)*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; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 6 - (2*n^3 + 3*n^2 + 25*n - 102)*x + (3*n^3 - 19*n^2 - 24*n + 40)*x^2 + (2*n^3 - 4*n^2 - 2*n + 4)*x^3)*v[i - 1] + (24*x - (4*n^2 + 8*n + 82)*x^2 + (7*n^2 - 10*n - 38)*x^3 + (2*n^2 - 2*n - 4)*x^4)*deriv(v[i - 1], x))/(12*n + 6 - (2*n^3 + 3*n^2 + 37*n - 42)*x + (4*n^3 - 12*n^2 + 8*n)*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-9e187d16084c6bf9
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-9e187d16084c6bf9
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-9e187d16084c6bf9
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-9e187d16084c6bf9

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-9e187d16084c6bf9

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8f881604bf5a9d75 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3cffc405672d7710 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023