A132393 → 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

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

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

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	3														4
2	2	9	9													20
3	6	33	54	27												120
4	24	150	315	270	81											840
5	120	822	2025	2295	1215	243										6720
6	720	5292	14616	19845	14175	5103	729									60480
7	5040	39204	118188	182763	158760	78246	20412	2187								604800
8	40320	328752	1063116	1816668	1818369	1102248	398034	78732	6561							6652800
9	362880	3079728	10554300	19539360	21815325	15375339	6889050	1902690	295245	19683						79833600
10	3628800	31885920	114782184	227056500	276771330	219199365	115016517	39694050	8660520	1082565	59049					1037836800
11	39916800	361631520	1358261784	2841968052	3725654130	3241507005	1922779782	781684101	214347870	37889775	3897234	177147				14529715200
12	479001600	4459328640	17384035968	38178401976	53233753716	50075046450	32797878399	15148548558	4917226743	1097720910	160436133	13817466	531441			217945728000
13	6227020800	59408277120	239370453504	548471333592	806574004236	810676864998	576597558537	295324766451	109369593333	29022052059	5378832459	660935457	48361131	1594323		3487131648000
14	87178291200	850396942080	3529411180416	8396710030800	12937450060080	13769198122680	10504396414512	5864339405925	2417148606015	734417508825	162369810603	25389593775	2659862205	167403915	4782969	59281238016000

Generated b-files: A001715 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-dabecbb1addaf41b · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-2a63385a6c1d1a3f

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 ?

A001715: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 3 ?

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,       3;
  2:      2,       9,        9;
  3:      6,      33,       54,       27;
  4:     24,     150,      315,      270,       81;
  5:    120,     822,     2025,     2295,     1215,      243;
  6:    720,    5292,    14616,    19845,    14175,     5103,     729;
  7:   5040,   39204,   118188,   182763,   158760,    78246,   20412,    2187;
  8:  40320,  328752,  1063116,  1816668,  1818369,  1102248,  398034,   78732,   6561;
  9: 362880, 3079728, 10554300, 19539360, 21815325, 15375339, 6889050, 1902690, 295245, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (n+3*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[(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; for(n = 1, N, my(i = n + 1); v[i] = (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)

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-3e76251141af3684
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-3e76251141af3684
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-3e76251141af3684

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-3e76251141af3684

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-edbf92ee2e446027 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.