A121574 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-2}(x) + \left(7 + 4\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 7 + 4\,x\)

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	4														11
2	54	60	16													130
3	413	684	368	64												1529
4	3161	6956	5632	1984	256											17989
5	24192	66408	71824	38208	9984	1024										211640
6	185149	609048	824384	587200	231936	48128	4096									2489941
7	1417003	5432740	8831632	7886272	4175104	1309696	225280	16384								29294111
8	10844766	47483028	90110496	96763968	64279296	27036672	7028736	1032192	65536							344644690
9	82998377	408591972	886594704	1112547648	889431552	469622784	163713024	36323328	4653056	262144						4054738589
10	635212469	3472931516	8481015408	12178474176	11384663808	7237386240	3167772672	942391296	182321152	20709376	1048576					47703926689
11	4861479168	29226323856	79326175328	128282497920	137303891712	102106199040	53821009920	20104298496	5214371840	894173184	91226112	4194304				561235840880
12	37206416521	243935691120	728485325824	1310098619264	1579894449408	1345684546560	829960273920	373397176320	121598967808	27949531136	4303355904	398458880	16777216			6602929589881
13	284752311487	2021953039876	6588676217312	13043348829056	17499305073152	16799116185600	11869989949440	6249426862080	2451270533120	707370942464	145954439168	20388511744	1728053248	67108864		77683348056611
14	2179298263014	16661184646764	58826715074288	127122581072256	187808397552384	200639034122240	159818933944320	96412774809600	44258184724480	15382822256640	3984479748096	745743056896	95328141312	7449083904	268435456	913943194931650

Technical details

Node id: n-21551e8b0efd7b64 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-bdd2ae59e2c72e1b

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,         4;
  2:       54,        60,        16;
  3:      413,       684,       368,         64;
  4:     3161,      6956,      5632,       1984,       256;
  5:    24192,     66408,     71824,      38208,      9984,      1024;
  6:   185149,    609048,    824384,     587200,    231936,     48128,      4096;
  7:  1417003,   5432740,   8831632,    7886272,   4175104,   1309696,    225280,    16384;
  8: 10844766,  47483028,  90110496,   96763968,  64279296,  27036672,   7028736,  1032192,   65536;
  9: 82998377, 408591972, 886594704, 1112547648, 889431552, 469622784, 163713024, 36323328, 4653056, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (5+4*x)*P(n-2,x) + (7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[(5 + 4*x)*P[n - 2, x] + (7 + 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (5 + 4*x)*v[i - 2] + (7 + 4*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_A013611 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    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 reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ 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/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

A121574 → basis A013611 → x ↦ x+1 15 rows; no all-index equality certificate was found

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