A119308 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n-3}(x) + \left(1 + 5\,x + 9\,x^{2} + 7\,x^{3} + 2\,x^{4} - 3\,n - 12\,n x - 19\,n x^{2} - 14\,n x^{3} - 4\,n x^{4}\right)\,P_{n-2}(x) + \left(4 + 15\,x + 18\,x^{2} + 7\,x^{3} + 3\,n + 11\,n x + 13\,n x^{2} + 5\,n x^{3}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

\(P_{2}(x) = 3 + 11\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	3	11	9													23
3	4	26	49	28												107
4	5	50	160	204	90											509
5	6	85	405	855	825	297										2473
6	7	133	875	2695	4235	3289	1001									12235
7	8	196	1694	7084	16016	20020	13013	3432								61463
8	9	276	3024	16380	49686	88452	91728	51272	11934							312761
9	10	375	5070	34398	133770	316680	464100	411060	201552	41990						1609005
10	11	495	8085	66990	323400	973896	1884960	2344980	1812030	791350	149226					8355423
11	12	638	12375	122760	718080	2665872	6523308	10659000	11511720	7887660	3105322	534888				43741635
12	13	806	18304	213928	1487772	6651216	19953648	41015832	57921006	55242044	33995104	12183560	1931540			230614773
13	14	1001	26299	357357	2909907	15380937	55288233	138705567	245402157	304893589	260275015	145348385	47805615	7020405		1223414481
14	15	1225	36855	575757	5420415	33378345	141216075	422751615	909431523	1410734325	1563837275	1207922625	617382675	187623765	25662825	6525975315

Technical details

Node id: n-cf87fca64596f51d · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-bed2a8b32a36c831

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1:  2,   3;
  2:  3,  11,    9;
  3:  4,  26,   49,    28;
  4:  5,  50,  160,   204,     90;
  5:  6,  85,  405,   855,    825,    297;
  6:  7, 133,  875,  2695,   4235,   3289,   1001;
  7:  8, 196, 1694,  7084,  16016,  20020,  13013,   3432;
  8:  9, 276, 3024, 16380,  49686,  88452,  91728,  51272,  11934;
  9: 10, 375, 5070, 34398, 133770, 316680, 464100, 411060, 201552, 41990;

%F Formula

The row polynomials satisfy (n+3+(2*n+6)*x+(n+3)*x^2)*P(n,x) = (n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n-3,x) - (3*n-1+(12*n-5)*x+(19*n-9)*x^2+(14*n-7)*x^3+(4*n-2)*x^4)*P(n-2,x) + (3*n+4+(11*n+15)*x+(13*n+18)*x^2+(5*n+7)*x^3)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 3 + 11*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 3 + 11*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*P[n - 3, x] - (3*n - 1 + (12*n - 5)*x + (19*n - 9)*x^2 + (14*n - 7)*x^3 + (4*n - 2)*x^4)*P[n - 2, x] + (3*n + 4 + (11*n + 15)*x + (13*n + 18)*x^2 + (5*n + 7)*x^3)*P[n - 1, x])/(n + 3 + (2*n + 6)*x + (n + 3)*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; v[2] = 2 + 3*x; v[3] = 3 + 11*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*v[i - 3] - (3*n - 1 + (12*n - 5)*x + (19*n - 9)*x^2 + (14*n - 7)*x^3 + (4*n - 2)*x^4)*v[i - 2] + (3*n + 4 + (11*n + 15)*x + (13*n + 18)*x^2 + (5*n + 7)*x^3)*v[i - 1])/(n + 3 + (2*n + 6)*x + (n + 3)*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle for first differences of Catalan numbers.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.