A075497 → 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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	3							5
2	4	14	11						29
3	8	52	92	49					201
4	16	184	556	644	257				1657
5	32	656	3024	5716	4854	1539			15821
6	64	2400	15856	43504	58976	39290	10299		170389
7	128	9024	82368	308176	593424	623288	340472	75905	2032785

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-276e0e2ea4eaffea · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-8e43e13b68ec28f7

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 ?

A004213: 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:   1;
  1:   2,      3;
  2:   4,     14,      11;
  3:   8,     52,      92,       49;
  4:  16,    184,     556,      644,      257;
  5:  32,    656,    3024,     5716,     4854,     1539;
  6:  64,   2400,   15856,    43504,    58976,    39290,     10299;
  7: 128,   9024,   82368,   308176,   593424,   623288,    340472,    75905;
  8: 256,  34688,  429248,  2110016,  5418736,  7981168,   6815656,  3147464,   609441;
  9: 512, 135424, 2259200, 14248256, 46926368, 90930064, 107937472, 77444632, 30919274, 5284451;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2-(2*n-7)*x-(4*n-7)*x^2-(2*n-2)*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*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[((2 - (2*n - 7)*x - (4*n - 7)*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + 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] = ((2 - (2*n - 7)*x - (4*n - 7)*x^2 - (2*n - 2)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + 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 [-1, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-8941cbe9a26788e8
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-8941cbe9a26788e8
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-8941cbe9a26788e8

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-8941cbe9a26788e8

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Stirling2 triangle with scaled diagonals (powers of 2).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-ca5f07049cdde3bf root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

No candidate equalities or related sequences.

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003