A064861 → basis A013610

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(6 + 21\,x + 18\,x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 6 + 21\,x + 18\,x^{2}\)

\(P_{1}(x) = 18 + 99\,x + 180\,x^{2} + 108\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	6	21	18															45
1	18	99	180	108														405
2	36	252	657	756	324													2025
3	108	972	3483	6210	5508	1944												18225
4	216	2268	9882	22869	29646	20412	5832											91125
5	648	8100	43254	127899	226152	239112	139968	34992										820125
6	1296	18144	110808	385560	836001	1156680	997272	489888	104976									4100625
7	3888	62208	441288	1821528	4821363	8486046	9931896	7453296	3254256	629856								36905625
8	7776	136080	1069200	4966920	15107310	31436181	45321930	44702280	28868400	11022480	1889568							184528125
9	23328	454896	4024080	21315960	75123450	184952403	324582876	406038420	354818880	206277840	71803584	11337408						1660753125
10	46656	979776	9412848	54704160	214194780	595275156	1204022961	1785825468	1927753020	1477012320	762440688	238085568	34012224					8303765625
11	139968	3219264	34117200	220589568	970809300	3070994148	7183719819	12581614170	16498211868	15997555080	11149395984	5288900832	1530550080	204073344				74733890625
12	279936	6858432	77892192	543530736	2603387304	9054416196	23580422082	46714387797	70741266246	81489745764	70291457208	44025989616	18927802656	4999796928	612220032			373669453125
13	839808	22254912	274827168	2097945360	11071346328	42783572412	125067763422	281625695883	492510125520	668916834768	699812846208	553826712096	320939345664	128566206720	31835441664	3673320192		3363025078125
14	1679616	47029248	616419072	5020372224	28436528736	118781183808	378486244080	938454682032	1829897338689	2815364046096	3406376196720	3207091962816	2303358827616	1219950450432	449369503488	102852965376	11019960576	16815125390625

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Technical details

Node id: n-0d95f7d056b9311c · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-2e6f63c0c32e8505

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     6,     21,      18;
  1:    18,     99,     180,      108;
  2:    36,    252,     657,      756,      324;
  3:   108,    972,    3483,     6210,     5508,      1944;
  4:   216,   2268,    9882,    22869,    29646,     20412,      5832;
  5:   648,   8100,   43254,   127899,   226152,    239112,    139968,     34992;
  6:  1296,  18144,  110808,   385560,   836001,   1156680,    997272,    489888,    104976;
  7:  3888,  62208,  441288,  1821528,  4821363,   8486046,   9931896,   7453296,   3254256,    629856;
  8:  7776, 136080, 1069200,  4966920, 15107310,  31436181,  45321930,  44702280,  28868400,  11022480,  1889568;
  9: 23328, 454896, 4024080, 21315960, 75123450, 184952403, 324582876, 406038420, 354818880, 206277840, 71803584, 11337408;

%F Formula

The row polynomials satisfy P(n,x) = (6+21*x+18*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 6 + 21*x + 18*x^2, P(1,x) = 18 + 99*x + 180*x^2 + 108*x^3.

%t Mathematica

P[0, x_] := 6 + 21*x + 18*x^2;
P[1, x_] := 18 + 99*x + 180*x^2 + 108*x^3;
P[n_, x_] := P[n, x] = Expand[(6 + 21*x + 18*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 21*x + 18*x^2; v[2] = 18 + 99*x + 180*x^2 + 108*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 21*x + 18*x^2)*v[i - 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-f5362ff16cbef54d
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-f5362ff16cbef54d
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-f5362ff16cbef54d

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A013610 e-f5362ff16cbef54d

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-bbffc3f85cdc52e2 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.