A122542 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-2}(x) + \left(5 + 3\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	3														7
3	24	30	9													63
4	136	246	144	27												553
5	776	1830	1584	594	81											4865
6	4424	12870	14724	8262	2268	243										42791
7	25224	87270	124056	92610	38232	8262	729									376383
8	143816	576774	979596	912438	502848	163782	29160	2187								3310601
9	819976	3740070	7386336	8243586	5682312	2475198	664848	100602	6561							29119489
10	4675144	23898822	53800596	69965478	57891024	31586598	11357820	2593782	341172	19683						256130119
11	26655624	150939750	381465000	566362530	546811560	358553790	161633880	49439322	9815256	1141614	59049					2252877375
12	151978696	944286342	2647043100	4417471350	4874605920	3733223094	2024021844	776546838	206540280	36256086	3779136	177147				19815929833
13	866515976	5861093670	18046753776	33438331170	41511777480	36364583070	23041975392	10637458650	3549920904	834913494	131324976	12400290	531441			174297580289
14	4940494664	36138097734	121238081316	246942931878	340624718640	335914957926	243599382828	131491472310	52817782104	15588975366	3285250164	468022374	40389516	1594323		1533092151143
15	28168537224	221555895270	804374995896	1786322489346	2710314492264	2971442610270	2427003438696	1499931270810	704675387016	250387639830	66223217304	12639438450	1645341336	130734486	4782969	13484820271167

Technical details

Node id: n-d243e2369f6dec16 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-72258eb2c4104da3

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       1;
   2:       4,        3;
   3:      24,       30,        9;
   4:     136,      246,      144,       27;
   5:     776,     1830,     1584,      594,       81;
   6:    4424,    12870,    14724,     8262,     2268,      243;
   7:   25224,    87270,   124056,    92610,    38232,     8262,      729;
   8:  143816,   576774,   979596,   912438,   502848,   163782,    29160,    2187;
   9:  819976,  3740070,  7386336,  8243586,  5682312,  2475198,   664848,  100602,   6561;
  10: 4675144, 23898822, 53800596, 69965478, 57891024, 31586598, 11357820, 2593782, 341172, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (4+3*x)*P(n-2,x) + (5+3*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(4 + 3*x)*P[n - 2, x] + (5 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 3*x; for(n = 3, N, my(i = n); v[i] = (4 + 3*x)*v[i - 2] + (5 + 3*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: 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 ≥ 1, 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 ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ 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 basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - ↑ Geometric basis map (elementary) (above)
  - ↑ Geometric basis map (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