A200536 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 11\,x + 2\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	15	11	2													28
2	225	330	181	44	4											784
3	3375	7425	6795	3311	906	132	8									21952
4	50625	148500	190350	139260	63601	18568	3384	352	16							614656
5	759375	2784375	4590000	4479750	2866575	1256651	382210	79640	10880	880	32					17210368
6	11390625	50118750	100996875	123255000	101455875	59341590	25289461	7912212	1803660	292160	31920	2112	64			481890304
7	170859375	877078125	2089040625	3060028125	3079636875	2252648475	1235011155	515550431	164668154	40047084	7299880	967120	88032	4928	128	13492928512

All 15 rows

Technical details

Node id: n-f2085d6c0d79c3b9 · revision 0 · JSON
Definition source: derived:shift_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:          15,           11,            2;
  2:         225,          330,          181,            44,             4;
  3:        3375,         7425,         6795,          3311,           906,           132,             8;
  4:       50625,       148500,       190350,        139260,         63601,         18568,          3384,           352,           16;
  5:      759375,      2784375,      4590000,       4479750,       2866575,       1256651,        382210,         79640,        10880,          880,          32;
  6:    11390625,     50118750,    100996875,     123255000,     101455875,      59341590,      25289461,       7912212,      1803660,       292160,       31920,        2112,         64;
  7:   170859375,    877078125,   2089040625,    3060028125,    3079636875,    2252648475,    1235011155,     515550431,    164668154,     40047084,     7299880,      967120,      88032,      4928,      128;
  8:  2562890625,  15035625000,  41325187500,   70634025000,   84032943750,   73785789000,   49463574300,   25823676120,  10611099361,   3443156816,   879352432,   174899648,   26558560,   2976512,   232192,   11264,    256;
  9: 38443359375, 253726171875, 790395468750, 1544158687500, 2120118806250, 2172417266250, 1721663181000, 1079026037100, 542154076335, 220016797451, 72287210178, 19182685104, 4080979392, 686591136, 89341632, 8676096, 592128, 25344, 512;

%F Formula

The row polynomials satisfy P(n,x) = (15+11*x+2*x^2)*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[(15 + 11*x + 2*x^2)*P[n - 1, x]];
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] = (15 + 11*x + 2*x^2)*v[i - 1]); 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]
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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A007318 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
  - Affine, reversal and Möbius substitutions (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 (above)
    
    ↑ Iterated real-rootedness preservers (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A007318 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) (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A200536 → basis A013610 → x ↦ x/(x+1)

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.