A200536 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 21\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	18	21	6													45
2	324	756	657	252	36											2025
3	5832	20412	29646	22869	9882	2268	216									91125
4	104976	489888	997272	1156680	836001	385560	110808	18144	1296							4100625
5	1889568	11022480	28868400	44702280	45321930	31436181	15107310	4966920	1069200	136080	7776					184528125
6	34012224	238085568	762440688	1477012320	1927753020	1785825468	1204022961	595275156	214194780	54704160	9412848	979776	46656			8303765625
7	612220032	4999796928	18927802656	44025989616	70291457208	81489745764	70741266246	46714387797	23580422082	9054416196	2603387304	543530736	77892192	6858432	279936	373669453125

All 15 rows

Technical details

Node id: n-e2e944875f74226f · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-10e88990a58171dc

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:           18,            21,              6;
  2:          324,           756,            657,            252,             36;
  3:         5832,         20412,          29646,          22869,           9882,            2268,             216;
  4:       104976,        489888,         997272,        1156680,         836001,          385560,          110808,           18144,            1296;
  5:      1889568,      11022480,       28868400,       44702280,       45321930,        31436181,        15107310,         4966920,         1069200,         136080,           7776;
  6:     34012224,     238085568,      762440688,     1477012320,     1927753020,      1785825468,      1204022961,       595275156,       214194780,       54704160,        9412848,         979776,         46656;
  7:    612220032,    4999796928,    18927802656,    44025989616,    70291457208,     81489745764,     70741266246,     46714387797,     23580422082,     9054416196,     2603387304,      543530736,      77892192,       6858432,       279936;
  8:  11019960576,  102852965376,   449369503488,  1219950450432,  2303358827616,   3207091962816,   3406376196720,   2815364046096,   1829897338689,   938454682032,   378486244080,   118781183808,   28436528736,    5020372224,    616419072,    47029248,    1679616;
  9: 198359290368, 2082772548864, 10314683099136, 32012985473280, 69775635377088, 113417893413216, 142483855725792, 141453004737744, 112499054244738, 72212212665621, 37499684748246, 15717000526416, 5277179841696, 1400220906336, 287142532416, 43913560320, 4716361728, 317447424, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (18+21*x+6*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[(18 + 21*x + 6*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] = (18 + 21*x + 6*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 (−∞, 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 ?

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.