basis-map coefficients of A371080, shifted → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

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

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

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	2														3
2	4	10	4													18
3	28	78	48	8												162
4	280	836	636	176	16											1944
5	3640	11428	9940	3560	560	32										29160
6	58240	190128	181896	76840	16080	1632	64									524880
7	1106560	3728912	3836280	1823752	459200	63168	4480	128								11022480
8	24344320	84249184	91855984	47795104	13749904	2308096	224896	11776	256							264539520
9	608608000	2154918240	2464897968	1378589568	439337808	85202208	10238592	744192	29952	512						7142567040
10	17041024000	61554926720	73326979584	43530303840	15058637760	3264337440	457084992	41314560	2327040	74240	1024					214277011200
11	528271744000	1942284776320	2396246220544	1496093378208	553878378240	131311736160	20698309632	2194921344	154767360	6955520	180224	2048				7071141369600
12	17961239296000	67094225882880	85356941051136	55659667300160	21824051616576	5572355785920	966365999808	116023944960	9651932928	546022400	20038656	430080	4096			254561089305600
13	664565853952000	2518408836258560	3292395270657792	2230121572208192	918809244413632	249825267312192	46900253564736	6225617963136	589169408256	39506694656	1833475072	55990272	1011712	8192		9927882482918400
14	26582634158080000	102065485158246400	136732628498828800	95789653429643264	41212612920961664	11830629181314944	2375660677213824	342825225654912	36018012256512	2758606602752	152352392192	5906561024	152449024	2351104	16384	416971064282572800

Generated b-files: A032031 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-20bccc4c38847de1 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-d1d0e40f5b436a84

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 ?

A032031: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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,          2;
  2:         4,         10,          4;
  3:        28,         78,         48,          8;
  4:       280,        836,        636,        176,        16;
  5:      3640,      11428,       9940,       3560,       560,       32;
  6:     58240,     190128,     181896,      76840,     16080,     1632,       64;
  7:   1106560,    3728912,    3836280,    1823752,    459200,    63168,     4480,    128;
  8:  24344320,   84249184,   91855984,   47795104,  13749904,  2308096,   224896,  11776,   256;
  9: 608608000, 2154918240, 2464897968, 1378589568, 439337808, 85202208, 10238592, 744192, 29952, 512;

%F Formula

The row polynomials satisfy P(n,x) = (3*n-2+2*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 + 2*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 + 2*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-d3f9f2e05c32db01
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-d3f9f2e05c32db01
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-d3f9f2e05c32db01

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 A013609 e-d3f9f2e05c32db01

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.