A079510 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	6														8
2	9	45	54													108
3	44	336	828	648												1856
4	265	2700	10125	16470	9720											39280
5	1854	23778	120366	299862	366444	174960										987264
6	14833	229845	1469475	4954635	9276120	9124164	3674160									28743232
7	133496	2429472	18805680	80192160	203248440	305814528	252572256	88179840								951375872
8	1334961	27938736	254277954	1313750340	4211684505	8572611348	10809562428	7712271792	2380855680							35284287744
9	14684570	347735790	3641882580	22131934020	85967604450	221232247110	376974123120	409868556840	257822105760	71425670400						1449426544640
10	176214841	4661064309	55255025232	386475825582	1765679762385	5503769978553	11849180870142	17390250738036	16642399920696	9372380101920	2357047123200					65327276624896
11	2290792932	66977979984	887079697308	7023581796456	36929647747692	135359331914016	352813940043012	653743011295944	843606958933008	721733662220256	368261573281920	84853696435200				3205281752137728
12	32071101049	1027520878956	15044274120447	133081803265554	792031504890483	3340338097496400	10234281545681193	22946594122054746	37357438884423372	43054103545482744	33331177715831424	15557224249457280	3309294160972800			170071669495656448
13	481066515734	16766183572362	269007905524050	2630592251213850	17489882309538870	83476132422549174	294198628835376342	775062730267326270	1526028801407718300	2217305646328809960	2310377312771112816	1634365541385255168	703297391981103360	138990354760857600		9703509269876473856
14	7697064251745	289996759525005	5061499394976765	54237458901193425	398594039668013475	2124933179450004495	8473234829117913855	25669620832988304315	59361213201904270800	104257777061260765800	136871816101647335280	130215723556433177520	84845624745693202560	33882156197806291200	6254565964238592000	592414856362327818240

Technical details

Node id: n-9e70cba966d7e1d5 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-022c58b01aaed433

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:        2,         6;
  2:        9,        45,         54;
  3:       44,       336,        828,         648;
  4:      265,      2700,      10125,       16470,        9720;
  5:     1854,     23778,     120366,      299862,      366444,       174960;
  6:    14833,    229845,    1469475,     4954635,     9276120,      9124164,      3674160;
  7:   133496,   2429472,   18805680,    80192160,   203248440,    305814528,    252572256,     88179840;
  8:  1334961,  27938736,  254277954,  1313750340,  4211684505,   8572611348,  10809562428,   7712271792,   2380855680;
  9: 14684570, 347735790, 3641882580, 22131934020, 85967604450, 221232247110, 376974123120, 409868556840, 257822105760, 71425670400;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(3*n+3)*x)*P(n-2,x) + (n+1+(3*n+3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (3*n + 3)*x)*P[n - 2, x] + (n + 1 + (3*n + 3)*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] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (3*n + 3)*x)*v[i - 2] + (n + 1 + (3*n + 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 - 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 Triangle T(n,k) read by rows; related to number of preorders.?
  - 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