basis-map coefficients of A371080, shifted → basis A013611

nonnegative coefficients only rational roots

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

Recurrence

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

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

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	4														5
2	4	20	16													40
3	28	156	192	64												440
4	280	1672	2544	1408	256											6160
5	3640	22856	39760	28480	8960	1024										104720
6	58240	380256	727584	614720	257280	52224	4096									2094400
7	1106560	7457824	15345120	14590016	7347200	2021376	286720	16384								48171200
8	24344320	168498368	367423936	382360832	219998464	73859072	14393344	1507328	65536							1252451200
9	608608000	4309836480	9859591872	11028716544	7029404928	2726470656	655269888	95256576	7667712	262144						36321084800
10	17041024000	123109853440	293307918336	348242430720	240938204160	104458798080	29253439488	5288263680	595722240	38010880	1048576					1162274713600
11	528271744000	3884569552640	9584984882176	11968747025664	8862054051840	4201975557120	1324691816448	280949932032	39620444160	3561226240	184549376	4194304				40679614976000
12	17961239296000	134188451765760	341427764204544	445277338401280	349184825865216	178315385149440	61847423987712	14851064954880	2470894829568	279563468800	20519583744	880803840	16777216			1545825369088000
13	664565853952000	5036817672517120	13169581082631168	17840972577665536	14700947910618112	7994408553990144	3001616228143104	796879099281408	150827368513536	20227427663872	1877478473728	114668077056	4143972352	67108864		63378840132608000
14	26582634158080000	204130970316492800	546930513995315200	766317227437146112	659401806735386624	378580133802078208	152042283341684736	43881628883828736	9220611137667072	1412406580609024	156008849604608	12096636977152	624431202304	19260243968	268435456	2788668965834752000

Generated b-files: A034000 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-1a01d480c8d600c3 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-0c88df3e4716fde5

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): yes

OEIS matches ?

A034000: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:         1;
  1:         1,          4;
  2:         4,         20,         16;
  3:        28,        156,        192,          64;
  4:       280,       1672,       2544,        1408,        256;
  5:      3640,      22856,      39760,       28480,       8960,       1024;
  6:     58240,     380256,     727584,      614720,     257280,      52224,      4096;
  7:   1106560,    7457824,   15345120,    14590016,    7347200,    2021376,    286720,    16384;
  8:  24344320,  168498368,  367423936,   382360832,  219998464,   73859072,  14393344,  1507328,   65536;
  9: 608608000, 4309836480, 9859591872, 11028716544, 7029404928, 2726470656, 655269888, 95256576, 7667712, 262144;

%F Formula

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

%t Mathematica

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

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-c271a57d901f1e05
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-c271a57d901f1e05
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-c271a57d901f1e05

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 A013611 e-c271a57d901f1e05

Geometric basis map (elementary) polytool ?
- premise (sequence): basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-aa6a1f197b91b3b6 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.