A103633 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 42\,x + 36\,x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	12	42	36													90
3	36	198	360	216												810
4	144	1008	2628	3024	1296											8100
5	432	3888	13932	24840	22032	7776										72900
6	1728	18144	79056	182952	237168	163296	46656									729000
7	5184	64800	346032	1023192	1809216	1912896	1119744	279936								6561000
8	20736	290304	1772928	6168960	13376016	18506880	15956352	7838208	1679616							65610000
9	62208	995328	7060608	29144448	77141808	135776736	158910336	119252736	52068096	10077696						590490000
10	248832	4354560	34214400	158941440	483433920	1005957792	1450301760	1430472960	923788800	352719360	60466176					5904900000
11	746496	14556672	128770560	682110720	2403950400	5918476896	10386652032	12993229440	11354204160	6600890880	2297714688	362797056				53144100000
12	2985984	62705664	602422272	3501066240	13708465920	38097609984	77057469504	114292829952	123376193280	94528788480	48796204032	15237476352	2176782336			531441000000
13	8957952	206032896	2183500800	14117732352	62131795200	196543625472	459758068416	805223306880	1055885559552	1023843525120	713561342976	338489653248	97955205120	13060694016		4782969000000
14	35831808	877879296	9970200576	69571934208	333233574912	1158965273088	3018294026496	5979441638016	9054882079488	10430687457792	8997306522624	5635326670848	2422758739968	639974006784	78364164096	47829690000000

Technical details

Node id: n-ff6e5b4b9776bc17 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-2e6f63c0c32e8505

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:     3,      6;
  2:    12,     42,      36;
  3:    36,    198,     360,      216;
  4:   144,   1008,    2628,     3024,     1296;
  5:   432,   3888,   13932,    24840,    22032,      7776;
  6:  1728,  18144,   79056,   182952,   237168,    163296,     46656;
  7:  5184,  64800,  346032,  1023192,  1809216,   1912896,   1119744,    279936;
  8: 20736, 290304, 1772928,  6168960, 13376016,  18506880,  15956352,   7838208,  1679616;
  9: 62208, 995328, 7060608, 29144448, 77141808, 135776736, 158910336, 119252736, 52068096, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (12+42*x+36*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[(12 + 42*x + 36*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (12 + 42*x + 36*x^2)*v[i - 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 (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool 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.