A200536 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 45\,x + 117\,x^{2} + 123\,x^{3} + 45\,x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	33	45													84
2	36	396	1629	2970	2025											7056
3	216	3564	24462	89397	183465	200475	91125									592704
4	1296	28512	274104	1504008	5151681	11280060	15418350	12028500	4100625							49787136
5	7776	213840	2643840	19352520	92877030	305366193	696577725	1088579250	1115370000	676603125	184528125					4182119424
6	46656	1539648	23269680	212984640	1314868140	5768002548	18436017069	43260019110	73961332875	89852895000	73626721875	36536568750	8303765625			351298031616
7	279936	10777536	192525984	2115091440	15964837560	87582972708	360129252798	1127508792597	2700969395985	4926542214825	6735165845625	6692281509375	4568731846875	1918169859375	373669453125	29509034655744

All 15 rows

Technical details

Node id: n-f03e5c22b4dd0de2 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-8e69101448399e5f

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:        1;
  1:        6,        33,          45;
  2:       36,       396,        1629,         2970,          2025;
  3:      216,      3564,       24462,        89397,        183465,         200475,          91125;
  4:     1296,     28512,      274104,      1504008,       5151681,       11280060,       15418350,        12028500,          4100625;
  5:     7776,    213840,     2643840,     19352520,      92877030,      305366193,      696577725,      1088579250,       1115370000,        676603125,         184528125;
  6:    46656,   1539648,    23269680,    212984640,    1314868140,     5768002548,    18436017069,     43260019110,      73961332875,      89852895000,       73626721875,       36536568750,        8303765625;
  7:   279936,  10777536,   192525984,   2115091440,   15964837560,    87582972708,   360129252798,   1127508792597,    2700969395985,    4926542214825,     6735165845625,     6692281509375,     4568731846875,     1918169859375,      373669453125;
  8:  1679616,  73903104,  1523411712,  19528895232,  174250712160,  1147516590528,  5769431306352,  22590551869776,   69619422907521,  169429139023320,   324530510982300,   484108561629000,   551340143943750,   463429838025000,   271134555187500,    98648735625000,    16815125390625;
  9: 10077696, 498845952, 11654855424, 170771597568, 1758511342656, 13514173329888, 80325917372736, 377572790902032, 1422829157933574, 4330590624228033, 10671218684501805, 21238469488239300, 33887496391623000, 42759689051598750, 41730298463418750, 30393675446062500, 15557354011406250, 4994092241015625, 756680642578125;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+45*x+117*x^2+123*x^3+45*x^4)*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[((6 + 45*x + 117*x^2 + 123*x^3 + 45*x^4)*P[n - 1, x])/(1 + 2*x + 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; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 45*x + 117*x^2 + 123*x^3 + 45*x^4)*v[i - 1])/(1 + 2*x + 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 basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A200536 → basis A007318 → basis A007318

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.