A116412 → basis A013610 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 9\,x\right)\,P_{n-2}(x) + \left(6 + 9\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 7 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	9														16
2	46	126	81													253
3	304	1269	1701	729												4003
4	2008	11268	23085	20412	6561											63334
5	13264	93492	258147	348462	229635	59049										1002049
6	87616	743472	2584062	4703508	4723920	2480058	531441									15854077
7	578752	5742720	24069636	55194777	74729790	59698539	26040609	4782969								250837792
8	3822976	43427520	213129792	589865976	1006358985	1083194856	717976791	267846264	43046721							3968669881
9	25252864	323151552	1817589456	5894769816	12142619847	16467762267	14698063737	8322366060	2711943423	387420489						62790939511
10	166809088	2374301952	15057267552	56004556032	135242877150	221480162514	249018903693	189749946168	93755758338	27119434230	3486784401					993456801118
11	1101865984	17266975488	121891044672	511480128528	1417121674920	2721221497125	3694437000135	3545241848118	2356033133772	1032475603185	268482398877	31381059609				15718134230413
12	7278432256	124517136384	968346835200	4525433805312	14151063719160	31184530601436	49647012552333	57521554007832	48126147984558	28351431385020	11161196867601	2636009007156	282429536481			248686951870729
13	48078057472	891593404416	7573702196736	39010664264832	135907094419200	338105738143836	617811592201587	839383337411676	847782183145986	628578120787830	332496273694959	118808691679674	25701087819771	2541865828329		3934640023056304
14	317582073856	6345837379584	58460595388416	329044162097664	1263871704023136	3503895974514000	7229070022126302	11273513683286664	13351341713589720	11948049430441956	7951988399195808	3816313421169708	1248338551246020	249102851176242	22876792454961	62252530720164037

Technical details

Node id: n-3495cde1f187c7d1 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-cd29e65288cea412

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:        1;
  1:        7,         9;
  2:       46,       126,         81;
  3:      304,      1269,       1701,        729;
  4:     2008,     11268,      23085,      20412,        6561;
  5:    13264,     93492,     258147,     348462,      229635,       59049;
  6:    87616,    743472,    2584062,    4703508,     4723920,     2480058,      531441;
  7:   578752,   5742720,   24069636,   55194777,    74729790,    59698539,    26040609,    4782969;
  8:  3822976,  43427520,  213129792,  589865976,  1006358985,  1083194856,   717976791,  267846264,   43046721;
  9: 25252864, 323151552, 1817589456, 5894769816, 12142619847, 16467762267, 14698063737, 8322366060, 2711943423, 387420489;

%F Formula

The row polynomials satisfy P(n,x) = (4+9*x)*P(n-2,x) + (6+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 9*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + 9*x;
P[n_, x_] := P[n, x] = Expand[(4 + 9*x)*P[n - 2, x] + (6 + 9*x)*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; v[2] = 7 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 9*x)*v[i - 2] + (6 + 9*x)*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 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003