A272867 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(33 + 36\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	33	36	9													78
2	1089	2376	1890	648	81											6084
3	35937	117612	157707	110808	43011	8748	729									474552
4	1185921	5174928	9761796	10392624	6827814	2834352	726084	104976	6561							37015056
5	39135393	213465780	519109965	740955600	687308490	432868536	187447770	55112400	10530405	1180980	59049					2887174368
6	1291467969	8453244888	25167615462	45060685560	54027571455	45696367728	27954820116	12462645744	4018579695	914078520	139237542	12754584	531441			225199600704
7	42618442977	325449928188	1146471337935	2469115984104	3631603077333	3858518877444	3053826445131	1828908143280	832861757763	286997271876	73668882861	13660159464	1729840455	133923132	4782969	17565568854912

All 15 rows

Technical details

Node id: n-67a731b2b62168b7 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-38e8cf17451719bf

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:              1;
  1:             33,              36,                9;
  2:           1089,            2376,             1890,              648,                81;
  3:          35937,          117612,           157707,           110808,             43011,              8748,               729;
  4:        1185921,         5174928,          9761796,         10392624,           6827814,           2834352,            726084,            104976,              6561;
  5:       39135393,       213465780,        519109965,        740955600,         687308490,         432868536,         187447770,          55112400,          10530405,          1180980,            59049;
  6:     1291467969,      8453244888,      25167615462,      45060685560,       54027571455,       45696367728,       27954820116,       12462645744,        4018579695,        914078520,        139237542,         12754584,          531441;
  7:    42618442977,    325449928188,    1146471337935,    2469115984104,     3631603077333,     3858518877444,     3053826445131,     1828908143280,      832861757763,     286997271876,      73668882861,      13660159464,      1729840455,       133923132,        4782969;
  8:  1406408618241,  12274111577376,   49933317553416,  125682844994784,   219049319021148,   280290877596576,   272367379973304,   205018390649952,   120809569170438,   55914106540896,   20258730741816,    5685840492192,   1211870421468,    189635154912,    20547634824,    1377495072,     43046721;
  9: 46411484401953, 455676392310084, 2102325173612433, 6055600320947232, 12202609805490852, 18266520050401392, 21050039003786100, 19093450468856544, 13818684265782462, 8039475521834904, 3768732072486126, 1420174001815776, 427010558303700, 101057859714672, 18411771319236, 2491888585248, 235939077801, 13947137604, 387420489;

%F Formula

The row polynomials satisfy P(n,x) = (33+36*x+9*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[(33 + 36*x + 9*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] = (33 + 36*x + 9*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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    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, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

A272867 → basis A013609 → 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.