A122542 → basis A013610 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 12\,x\right)\,P_{n-2}(x) + \left(5 + 12\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 4 + 12\,x\)

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	12														16
3	24	120	144													288
4	136	984	2304	1728												5152
5	776	7320	25344	38016	20736											92192
6	4424	51480	235584	528768	580608	248832										1649696
7	25224	349080	1984896	5927040	9787392	8460288	2985984									29519904
8	143816	2307096	15673536	58396032	128729088	167712768	119439360	35831808								528233504
9	819976	14960280	118181376	527589504	1454671872	2534602752	2723217408	1648263168	429981696							9452288032
10	4675144	95595288	860809536	4477790592	14820102144	32344676352	46521630720	42496524288	22359048192	5159780352						169140632608
11	26655624	603759000	6103440000	36247201920	139983759360	367159080960	662052372480	810013851648	643252617216	299267260416	61917364224					3026627362848
12	151978696	3777145368	42352689600	282718166400	1247899115520	3822820448256	8290393473024	12722943393792	13535823790080	9504315408384	3962711310336	743008370688				54158915290144
13	866515976	23444374680	288748060416	2140053194880	10627015034880	37237333063680	94379931205632	174284122521600	232647616364544	218867562971136	137704218034176	52010585948160	8916100448256			969127597738016
14	4940494664	144552390936	1939809301056	15804347640192	87199927971840	343976916916224	997783072063488	2154356282327040	3461466167967744	4086556358344704	3444834475966464	1963028115357696	677623634067456	106993205379072		17341711806188576
15	28168537224	886223581080	12869999934336	114324639318144	693840510019584	3042757232916480	9941006084898816	24574873940951040	46181606163480576	65637617455595520	69440076307759104	53013647248588800	27604246987800576	8773442841083904	1283918464548864	310315142269014048

Technical details

Node id: n-35819491ecf79d70 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-dda21e41e51cc759

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,       12;
   3:      24,      120,       144;
   4:     136,      984,      2304,       1728;
   5:     776,     7320,     25344,      38016,       20736;
   6:    4424,    51480,    235584,     528768,      580608,      248832;
   7:   25224,   349080,   1984896,    5927040,     9787392,     8460288,     2985984;
   8:  143816,  2307096,  15673536,   58396032,   128729088,   167712768,   119439360,    35831808;
   9:  819976, 14960280, 118181376,  527589504,  1454671872,  2534602752,  2723217408,  1648263168,   429981696;
  10: 4675144, 95595288, 860809536, 4477790592, 14820102144, 32344676352, 46521630720, 42496524288, 22359048192, 5159780352;

%F Formula

The row polynomials satisfy P(n,x) = (4+12*x)*P(n-2,x) + (5+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 12*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[(4 + 12*x)*P[n - 2, x] + (5 + 12*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 + 12*x; for(n = 3, N, my(i = n); v[i] = (4 + 12*x)*v[i - 2] + (5 + 12*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 (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 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)

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