A082985 → 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

\(\left(-2\,n - 2\,n x + 4\,n^{2} + 8\,n^{3} x\right)\,P_{n}(x) = \left(-2\,x^{2} - 2\,n - 14\,n x - 18\,n x^{2} + 4\,n^{2} + 24\,n^{2} x + 32\,n^{2} x^{2} + 8\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-x^{2} - 4\,x^{3} - 2\,n x - 16\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

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	6														7
2	1	10	15													26
3	1	14	42	28												85
4	1	18	81	120	45											265
5	1	22	132	308	275	66										804
6	1	26	195	624	910	546	91									2393
7	1	30	270	1100	2250	2268	980	120								7019
8	1	34	357	1768	4675	6732	4998	1632	153							20350
9	1	38	456	2660	8645	16302	17556	10032	2565	190						58445
10	1	42	567	3808	14700	34398	49049	41184	18711	3850	231					166541
11	1	46	690	5244	23460	65688	117208	131560	88803	32890	5566	276				471432
12	1	50	825	7000	35625	116280	249900	353600	321750	178750	55055	7800	325			1326961
13	1	54	972	9108	51975	193914	488376	837216	966654	729300	339768	88452	10647	378		3716815
14	1	58	1131	11600	73370	308154	890967	1798464	2529090	2435420	1550978	615264	137228	14210	435	10366370

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Technical details

Node id: n-39faef7b70b61607 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-122247bb6f6b155e

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,  6;
  2: 1, 10,  15;
  3: 1, 14,  42,   28;
  4: 1, 18,  81,  120,   45;
  5: 1, 22, 132,  308,  275,    66;
  6: 1, 26, 195,  624,  910,   546,    91;
  7: 1, 30, 270, 1100, 2250,  2268,   980,   120;
  8: 1, 34, 357, 1768, 4675,  6732,  4998,  1632,  153;
  9: 1, 38, 456, 2660, 8645, 16302, 17556, 10032, 2565, 190;

%F Formula

The row polynomials satisfy (4*n^2-2*n+(8*n^3-2*n)*x)*P(n,x) = (4*n^2-2*n+(8*n^3+24*n^2-14*n)*x+(24*n^3+32*n^2-18*n-2)*x^2)*P(n-1,x) - (2*n*x+(4*n^2+16*n+1)*x^2+(16*n^2+32*n+4)*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[((4*n^2 - 2*n + (8*n^3 + 24*n^2 - 14*n)*x + (24*n^3 + 32*n^2 - 18*n - 2)*x^2)*P[n - 1, x] - (2*n*x + (4*n^2 + 16*n + 1)*x^2 + (16*n^2 + 32*n + 4)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 2*n + (8*n^3 - 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] = ((4*n^2 - 2*n + (8*n^3 + 24*n^2 - 14*n)*x + (24*n^3 + 32*n^2 - 18*n - 2)*x^2)*v[i - 1] - (2*n*x + (4*n^2 + 16*n + 1)*x^2 + (16*n^2 + 32*n + 4)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 2*n + (8*n^3 - 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-d6813283d076dba7
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-d6813283d076dba7
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-d6813283d076dba7
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-d6813283d076dba7

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-d6813283d076dba7

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-4fa62233b528a616 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