A249252 → 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

\(\left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(-2 + 12\,x + 106\,x^{2} + 260\,x^{3} + 294\,x^{4} + 160\,x^{5} + 34\,x^{6} - 3\,n - 46\,n x - 209\,n x^{2} - 432\,n x^{3} - 457\,n x^{4} - 242\,n x^{5} - 51\,n x^{6} + 5\,n^{2} + 34\,n^{2} x + 103\,n^{2} x^{2} + 172\,n^{2} x^{3} + 163\,n^{2} x^{4} + 82\,n^{2} x^{5} + 17\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(22\,x + 174\,x^{2} + 544\,x^{3} + 876\,x^{4} + 774\,x^{5} + 358\,x^{6} + 68\,x^{7} - n - 20\,n x - 115\,n x^{2} - 314\,n x^{3} - 471\,n x^{4} - 400\,n x^{5} - 181\,n x^{6} - 34\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 19\,x^{2} + 97\,x^{3} + 235\,x^{4} + 315\,x^{5} + 241\,x^{6} + 99\,x^{7} + 17\,x^{8}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 1 + 5\,x\)

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	5														6
2	1	10	17													28
3	1	15	59	61												136
4	1	20	126	292	217											656
5	1	25	218	818	1333	773										3168
6	1	30	335	1764	4623	5790	2753									15296
7	1	35	477	3255	11987	23961	24335	9805								73856
8	1	40	644	5416	25950	72696	117044	99896	34921							356608
9	1	45	836	8372	49662	181030	407028	547620	402889	124373						1721856
10	1	50	1053	12248	86898	392748	1147410	2148184	2479629	1602674	442961					8313856
11	1	55	1295	17169	142058	769510	2787038	6766530	10833477	10942579	6305507	1577629				40142848
12	1	60	1562	23260	220167	1393976	6054860	18207960	37724255	52698636	47297850	24585420	5618809			193826816
13	1	65	1854	30646	326875	2372931	12059924	43485636	111455271	201071415	248945870	200975142	95141341	20011685		935878656
14	1	70	2171	39452	468457	3840410	22407003	94563080	290437819	647301658	1033005897	1147805724	841833307	365844166	71272673	4518821888

Computed in your browser ?

Technical details

Node id: n-fb6c374737b17be6 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-78b8e1ecee5b9454

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 ?

A108051: 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: 1,  5;
  2: 1, 10,  17;
  3: 1, 15,  59,   61;
  4: 1, 20, 126,  292,   217;
  5: 1, 25, 218,  818,  1333,    773;
  6: 1, 30, 335, 1764,  4623,   5790,   2753;
  7: 1, 35, 477, 3255, 11987,  23961,  24335,   9805;
  8: 1, 40, 644, 5416, 25950,  72696, 117044,  99896,  34921;
  9: 1, 45, 836, 8372, 49662, 181030, 407028, 547620, 402889, 124373;

%F Formula

The row polynomials satisfy (2*n-2+(6*n-6)*x+(6*n-6)*x^2+(2*n-2)*x^3)*P(n,x) = (5*n^2-3*n-2+(34*n^2-46*n+12)*x+(103*n^2-209*n+106)*x^2+(172*n^2-432*n+260)*x^3+(163*n^2-457*n+294)*x^4+(82*n^2-242*n+160)*x^5+(17*n^2-51*n+34)*x^6)*P(n-1,x) - (n+(20*n-22)*x+(115*n-174)*x^2+(314*n-544)*x^3+(471*n-876)*x^4+(400*n-774)*x^5+(181*n-358)*x^6+(34*n-68)*x^7)*d/dx P(n-1,x) + (x+19*x^2+97*x^3+235*x^4+315*x^5+241*x^6+99*x^7+17*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 5*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 3*n - 2 + (34*n^2 - 46*n + 12)*x + (103*n^2 - 209*n + 106)*x^2 + (172*n^2 - 432*n + 260)*x^3 + (163*n^2 - 457*n + 294)*x^4 + (82*n^2 - 242*n + 160)*x^5 + (17*n^2 - 51*n + 34)*x^6)*P[n - 1, x] - (n + (20*n - 22)*x + (115*n - 174)*x^2 + (314*n - 544)*x^3 + (471*n - 876)*x^4 + (400*n - 774)*x^5 + (181*n - 358)*x^6 + (34*n - 68)*x^7)*D[P[n - 1, x], x] + (x + 19*x^2 + 97*x^3 + 235*x^4 + 315*x^5 + 241*x^6 + 99*x^7 + 17*x^8)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n^2 - 3*n - 2 + (34*n^2 - 46*n + 12)*x + (103*n^2 - 209*n + 106)*x^2 + (172*n^2 - 432*n + 260)*x^3 + (163*n^2 - 457*n + 294)*x^4 + (82*n^2 - 242*n + 160)*x^5 + (17*n^2 - 51*n + 34)*x^6)*v[i - 1] - (n + (20*n - 22)*x + (115*n - 174)*x^2 + (314*n - 544)*x^3 + (471*n - 876)*x^4 + (400*n - 774)*x^5 + (181*n - 358)*x^6 + (34*n - 68)*x^7)*deriv(v[i - 1], x) + (x + 19*x^2 + 97*x^3 + 235*x^4 + 315*x^5 + 241*x^6 + 99*x^7 + 17*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)); 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-b4c8d04a96ca0f68
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-b4c8d04a96ca0f68
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-b4c8d04a96ca0f68

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-b4c8d04a96ca0f68

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-fb471901eb718e65 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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