A207606 → 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

\(\left(18 + 36\,x + 18\,x^{2} + 24\,n + 48\,n x + 24\,n x^{2} + 8\,n^{2} + 16\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-6 - 57\,x - 153\,x^{2} - 159\,x^{3} - 57\,x^{4} - 4\,n - 53\,n x - 144\,n x^{2} - 130\,n x^{3} - 20\,n x^{4} + 15\,n x^{5} + 5\,n^{2} + 15\,n^{2} x + 22\,n^{2} x^{2} + 34\,n^{2} x^{3} + 37\,n^{2} x^{4} + 15\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(18 + 123\,x + 318\,x^{2} + 396\,x^{3} + 240\,x^{4} + 57\,x^{5} - 8\,n x - 58\,n x^{2} - 156\,n x^{3} - 200\,n x^{4} - 124\,n x^{5} - 30\,n x^{6}\right)\,P_{n-1}'(x) + \left(12\,x + 75\,x^{2} + 195\,x^{3} + 270\,x^{4} + 210\,x^{5} + 87\,x^{6} + 15\,x^{7}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 4 + 15\,x + 13\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	4	15	13															32
1	5	31	58	34														128
2	6	54	162	201	89													512
3	7	85	361	707	655	233												2048
4	8	125	701	1918	2778	2052	610											8192
5	9	175	1239	4425	8880	10188	6255	1597										32768
6	10	236	2044	9122	23720	37490	35582	18687	4181									131072
7	11	309	3198	17298	55837	113767	148123	119825	54974	10946								524288
8	12	395	4797	30744	119560	300993	503265	556678	392286	159765	28657							2097152
9	13	495	6952	51876	237721	717927	1478015	2093313	2011859	1255669	459743	75025						8388608
10	14	610	9790	83875	445225	1578160	3885000	6756470	8294750	7045900	3946020	1312200	196418					33554432
11	15	741	13455	130845	793650	3246750	9352890	19400670	29182410	31603350	24046800	12212280	3719643	514229				134217728
12	16	889	18109	197990	1357070	6321965	20959485	50755380	90874140	120362060	116571080	80314820	37309288	10482351	1346269			536870912
13	17	1055	23933	291811	2239315	11750245	44244825	123036015	256677780	403977700	477792430	418433650	263372033	112726771	29391490	3524578		2147483648
14	18	1240	31128	420324	3582904	20984340	88779900	279810190	669119220	1225123020	1719348620	1836425550	1467535008	850153894	337343034	82048737	9227465	8589934592

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Technical details

Node id: n-a9766ae9536fc7a1 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-d049d83a04ac9840

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 ?

A081654: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A004171: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A081294: 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:  4,  15,   13;
  1:  5,  31,   58,    34;
  2:  6,  54,  162,   201,     89;
  3:  7,  85,  361,   707,    655,    233;
  4:  8, 125,  701,  1918,   2778,   2052,     610;
  5:  9, 175, 1239,  4425,   8880,  10188,    6255,    1597;
  6: 10, 236, 2044,  9122,  23720,  37490,   35582,   18687,    4181;
  7: 11, 309, 3198, 17298,  55837, 113767,  148123,  119825,   54974,   10946;
  8: 12, 395, 4797, 30744, 119560, 300993,  503265,  556678,  392286,  159765,  28657;
  9: 13, 495, 6952, 51876, 237721, 717927, 1478015, 2093313, 2011859, 1255669, 459743, 75025;

%F Formula

The row polynomials satisfy (8*n^2+24*n+18+(16*n^2+48*n+36)*x+(8*n^2+24*n+18)*x^2)*P(n,x) = (5*n^2-4*n-6+(15*n^2-53*n-57)*x+(22*n^2-144*n-153)*x^2+(34*n^2-130*n-159)*x^3+(37*n^2-20*n-57)*x^4+(15*n^2+15*n)*x^5)*P(n-1,x) + (18-(8*n-123)*x-(58*n-318)*x^2-(156*n-396)*x^3-(200*n-240)*x^4-(124*n-57)*x^5-30*n*x^6)*d/dx P(n-1,x) + (12*x+75*x^2+195*x^3+270*x^4+210*x^5+87*x^6+15*x^7)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 4 + 15*x + 13*x^2.

%t Mathematica

P[0, x_] := 4 + 15*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 4*n - 6 + (15*n^2 - 53*n - 57)*x + (22*n^2 - 144*n - 153)*x^2 + (34*n^2 - 130*n - 159)*x^3 + (37*n^2 - 20*n - 57)*x^4 + (15*n^2 + 15*n)*x^5)*P[n - 1, x] + (18 - (8*n - 123)*x - (58*n - 318)*x^2 - (156*n - 396)*x^3 - (200*n - 240)*x^4 - (124*n - 57)*x^5 - 30*n*x^6)*D[P[n - 1, x], x] + (12*x + 75*x^2 + 195*x^3 + 270*x^4 + 210*x^5 + 87*x^6 + 15*x^7)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18 + (16*n^2 + 48*n + 36)*x + (8*n^2 + 24*n + 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 15*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((5*n^2 - 4*n - 6 + (15*n^2 - 53*n - 57)*x + (22*n^2 - 144*n - 153)*x^2 + (34*n^2 - 130*n - 159)*x^3 + (37*n^2 - 20*n - 57)*x^4 + (15*n^2 + 15*n)*x^5)*v[i - 1] + (18 - (8*n - 123)*x - (58*n - 318)*x^2 - (156*n - 396)*x^3 - (200*n - 240)*x^4 - (124*n - 57)*x^5 - 30*n*x^6)*deriv(v[i - 1], x) + (12*x + 75*x^2 + 195*x^3 + 270*x^4 + 210*x^5 + 87*x^6 + 15*x^7)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18 + (16*n^2 + 48*n + 36)*x + (8*n^2 + 24*n + 18)*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-f3e51c43b80b3409
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-f3e51c43b80b3409
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-f3e51c43b80b3409

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-f3e51c43b80b3409

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0)
  - theorem_check three_term_recurrence e-3222de37fcca30b3 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