A153518 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-15 - 30\,x + 5\,n + 10\,n x\right)\,P_{n-2}(x) + \left(-6 - 6\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 10 + 10\,x\)

\(P_{2}(x) = 50 + 100\,x + 8\,x^{2}\)

\(P_{3}(x) = 250 + 750\,x + 516\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	10	10														20
2	50	100	8													158
3	250	750	516	16												1532
4	750	3000	3572	1144	32											8498
5	2750	13750	23224	14672	2512	64										56972
6	9250	55500	121808	117232	45968	5472	128									355358
7	32250	225750	608236	783680	485680	128320	11840	256								2276012
8	110750	886000	2832012	4588072	3940880	1715040	335680	25472	512							14434418
9	382750	3444750	12734704	24840928	27323104	16810240	5443840	841984	54528	1024						91877852
10	1319250	13192500	55378968	126411744	169913184	136250688	63336960	16055808	2050304	116224	2048					584027678
11	4552250	50074750	235263956	615133104	977674656	969609984	594496896	217433856	44904704	4883456	246784	4096				3714278492
12	15700750	188409000	979497252	2886642520	5299298880	6274954560	4807405440	2337510144	695486720	120660480	11432960	522240	8192			23617529138
13	54162750	704115750	4012879784	13160584624	27411587120	37773303360	34833304320	21321969408	8464855808	2105758720	314255360	26398720	1101824	16384		150184293932
14	186829250	2615609500	16215567328	58575113936	136507263088	214737542560	231999788160	172072152576	86426185472	28699398656	6103797760	798248960	60264448	2318336	32768	955000112798

Technical details

Node id: n-5fc6172728a1754c · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-2fb91d8b80545e88

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:      2;
  1:     10,      10;
  2:     50,     100,        8;
  3:    250,     750,      516,       16;
  4:    750,    3000,     3572,     1144,       32;
  5:   2750,   13750,    23224,    14672,     2512,       64;
  6:   9250,   55500,   121808,   117232,    45968,     5472,     128;
  7:  32250,  225750,   608236,   783680,   485680,   128320,   11840,    256;
  8: 110750,  886000,  2832012,  4588072,  3940880,  1715040,  335680,  25472,   512;
  9: 382750, 3444750, 12734704, 24840928, 27323104, 16810240, 5443840, 841984, 54528, 1024;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(10*n-30)*x)*P(n-2,x) + (2*n-6+(2*n-6)*x)*P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 10 + 10*x, P(2,x) = 50 + 100*x + 8*x^2, P(3,x) = 250 + 750*x + 516*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 10*x;
P[2, x_] := 50 + 100*x + 8*x^2;
P[3, x_] := 250 + 750*x + 516*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (10*n - 30)*x)*P[n - 2, x] + (2*n - 6 + (2*n - 6)*x)*P[n - 1, x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 10*x; v[3] = 50 + 100*x + 8*x^2; v[4] = 250 + 750*x + 516*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (10*n - 30)*x)*v[i - 2] + (2*n - 6 + (2*n - 6)*x)*v[i - 1])/(n - 3)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.?
  - 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

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