A107131 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	2	6	4													12
3	4	18	24	8												54
4	9	52	104	80	16											261
5	21	150	400	480	240	32										1323
6	51	432	1440	2360	1920	672	64									6939
7	127	1246	4984	10360	11760	6944	1792	128								37341
8	323	3600	16800	42336	61824	51968	23296	4608	256							205011
9	835	10422	55584	164640	294336	322560	209664	73728	11520	512						1143801
10	2188	30230	181380	617520	1307040	1770048	1518720	787200	222720	28160	1024					6466230
11	5798	87846	585640	2253240	5512320	8899968	9520896	6589440	2787840	647680	67584	2048				36960300
12	15511	255708	1875192	8045840	22342320	41918976	53822208	46835712	26737920	9405440	1824768	159744	4096			213243435
13	41835	745498	5963984	28233920	87747088	187675488	281478912	295865856	214046976	102575616	30457856	5005312	372736	8192		1240219269
14	113634	2176566	18863572	97668480	335903568	806886080	1385319936	1709465472	1502941440	919446528	375158784	95234048	13418496	860160	16384	7263473148

Technical details

Node id: n-bc12291099a8ab59 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-7f5e6b8bb14328be

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:   1,     2;
  2:   2,     6,     4;
  3:   4,    18,    24,      8;
  4:   9,    52,   104,     80,     16;
  5:  21,   150,   400,    480,    240,     32;
  6:  51,   432,  1440,   2360,   1920,    672,     64;
  7: 127,  1246,  4984,  10360,  11760,   6944,   1792,   128;
  8: 323,  3600, 16800,  42336,  61824,  51968,  23296,  4608,   256;
  9: 835, 10422, 55584, 164640, 294336, 322560, 209664, 73728, 11520, 512;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (3*n-3+(4*n-4)*x-(4*n-4)*x^2)*P(n-2,x) + (2*n+1+(4*n+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (4*n - 4)*x - (4*n - 4)*x^2)*P[n - 2, x] + (2*n + 1 + (4*n + 2)*x)*P[n - 1, x])/(n + 2)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (4*n - 4)*x - (4*n - 4)*x^2)*v[i - 2] + (2*n + 1 + (4*n + 2)*x)*v[i - 1])/(n + 2)); 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from A Motzkin related triangle.?
  - 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