A075504 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(9 + 10\,x - 8\,x^{2} - 9\,x^{3} - 9\,n x - 18\,n x^{2} - 9\,n x^{3}\right)\,P_{n-1}(x) + \left(9\,x + 27\,x^{2} + 27\,x^{3} + 9\,x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 81 + 189\,x + 109\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	81	189	109								379
1	729	2754	3375	1351							8209
2	6561	37179	74196	63189	19612						200737
3	59049	498636	1469664	2012931	1316304	333451					5690035
4	531441	6908733	28546911	55984284	57798441	30476844	6493069				186739723
5	4782969	100973790	562441725	1476165951	2133083889	1752530256	770626233	141264820			6941869633
6	43046721	1564030863	11350516878	38127880251	72155063331	82023253479	55799807148	21009965247	3376695763		285450259681
7	387420489	25483658832	234547233822	978956305839	2329026896475	3435051709881	3214456667820	1864880782659	613753603947	87799365343	12784343645107

All 15 rows

Technical details

Node id: n-b25768308db07696 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-1f4d1b1758c360f5

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:          81,           189,             109;
  1:         729,          2754,            3375,            1351;
  2:        6561,         37179,           74196,           63189,            19612;
  3:       59049,        498636,         1469664,         2012931,          1316304,           333451;
  4:      531441,       6908733,        28546911,        55984284,         57798441,         30476844,          6493069;
  5:     4782969,     100973790,       562441725,      1476165951,       2133083889,       1752530256,        770626233,        141264820;
  6:    43046721,    1564030863,     11350516878,     38127880251,      72155063331,      82023253479,      55799807148,      21009965247,       3376695763;
  7:   387420489,   25483658832,    234547233822,    978956305839,    2329026896475,    3435051709881,    3214456667820,    1864880782659,     613753603947,      87799365343;
  8:  3486784401,  431199004257,   4950760335489,  25198450390530,   73387477411326,  134958178714743,  162970542219024,  129544451505072,   65492154934812,   19138101881139,   2465959810690;
  9: 31381059609, 7482639324546, 106622380198179, 654383639952747, 2290822029200682, 5113139456702124, 7653802166734338, 7817257648334421, 5396969799068220, 2416226894493534, 634595586685092, 74353064138749;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (9-(9*n-10)*x-(18*n+8)*x^2-(9*n+9)*x^3)*P(n-1,x) + (9*x+27*x^2+27*x^3+9*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 81 + 189*x + 109*x^2.

%t Mathematica

P[0, x_] := 81 + 189*x + 109*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9 - (9*n - 10)*x - (18*n + 8)*x^2 - (9*n + 9)*x^3)*P[n - 1, x] + (9*x + 27*x^2 + 27*x^3 + 9*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] = 81 + 189*x + 109*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((9 - (9*n - 10)*x - (18*n + 8)*x^2 - (9*n + 9)*x^3)*v[i - 1] + (9*x + 27*x^2 + 27*x^3 + 9*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)

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 Stirling2 triangle with scaled diagonals (powers of 9).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.