A232535 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(3 + 15\,x + 27\,x^{2} + 21\,x^{3} + 6\,x^{4} - 9\,n - 38\,n x - 61\,n x^{2} - 44\,n x^{3} - 12\,n x^{4} + 7\,n^{2} + 28\,n^{2} x + 41\,n^{2} x^{2} + 26\,n^{2} x^{3} + 6\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-3 - 18\,x - 42\,x^{2} - 48\,x^{3} - 27\,x^{4} - 6\,x^{5} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x) + \left(-6\,x - 30\,x^{2} - 60\,x^{3} - 60\,x^{4} - 30\,x^{5} - 6\,x^{6}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	3							4
2	1	10	12						23
3	1	21	64	48					134
4	1	36	200	352	192				781
5	1	55	480	1456	1792	768			4552
6	1	78	980	4480	9216	8704	3072		26531
7	1	105	1792	11424	34560	53504	40960	12288	154634

Show all 15 stored rows

Generated b-files: A038723 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-df244c0ad53b2f3d · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-3908c67b5f7a1c29

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 ?

A038723: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,   3;
  2: 1,  10,   12;
  3: 1,  21,   64,    48;
  4: 1,  36,  200,   352,    192;
  5: 1,  55,  480,  1456,   1792,    768;
  6: 1,  78,  980,  4480,   9216,   8704,    3072;
  7: 1, 105, 1792, 11424,  34560,  53504,   40960,   12288;
  8: 1, 136, 3024, 25536, 105600, 236544,  292864,  188416,  49152;
  9: 1, 171, 4800, 51744, 278784, 841984, 1490944, 1536000, 851968, 196608;

%F Formula

The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = (7*n^2-9*n+3+(28*n^2-38*n+15)*x+(41*n^2-61*n+27)*x^2+(26*n^2-44*n+21)*x^3+(6*n^2-12*n+6)*x^4)*P(n-1,x) - (3-(2*n-18)*x-(6*n-42)*x^2-(6*n-48)*x^3-(2*n-27)*x^4+6*x^5)*d/dx P(n-1,x) - (6*x+30*x^2+60*x^3+60*x^4+30*x^5+6*x^6)*d^2/dx^2 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[((7*n^2 - 9*n + 3 + (28*n^2 - 38*n + 15)*x + (41*n^2 - 61*n + 27)*x^2 + (26*n^2 - 44*n + 21)*x^3 + (6*n^2 - 12*n + 6)*x^4)*P[n - 1, x] - (3 - (2*n - 18)*x - (6*n - 42)*x^2 - (6*n - 48)*x^3 - (2*n - 27)*x^4 + 6*x^5)*D[P[n - 1, x], x] - (6*x + 30*x^2 + 60*x^3 + 60*x^4 + 30*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^2 + 2*n^2*x + n^2*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] = ((7*n^2 - 9*n + 3 + (28*n^2 - 38*n + 15)*x + (41*n^2 - 61*n + 27)*x^2 + (26*n^2 - 44*n + 21)*x^3 + (6*n^2 - 12*n + 6)*x^4)*v[i - 1] - (3 - (2*n - 18)*x - (6*n - 42)*x^2 - (6*n - 48)*x^3 - (2*n - 27)*x^4 + 6*x^5)*deriv(v[i - 1], x) - (6*x + 30*x^2 + 60*x^3 + 60*x^4 + 30*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^2 + 2*n^2*x + n^2*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 [-1, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-df9504f6b81fedcd
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-df9504f6b81fedcd
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-df9504f6b81fedcd

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 mobius, shift 0 e-df9504f6b81fedcd

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/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-a75bd3d9d36c6f35 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023