A269944 → 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

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

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

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	2	12	16													30
3	10	68	128	64												270
4	100	720	1552	1152	256											3780
5	1700	12640	29264	25792	8960	1024										79380
6	44200	335440	811424	787648	336128	62464	4096									2381400
7	1635400	12588080	31364448	32388672	15587328	3655680	401408	16384								97637400
8	81770000	635945600	1618574720	1744891392	908921088	245133312	34693120	2424832	65536							5272419600
9	5315050000	41663544000	107751139200	119892239360	66059436288	19569349632	3235586048	296386560	13959168	262144						363796952400
10	435834100000	3437670808000	9002247590400	10262168184320	5896442733056	1868924414976	343595454464	37246042112	2330198016	77332480	1048576					31286537906400
11	44019244100000	348948088008000	922977689862400	1072487976977920	636589388775936	212347136844800	42178838560768	5136232071168	384334168064	17131372544	415236096	4194304				3285086480172000
12	5370347780200000	42747743713376000	113999070515244800	134535443950755840	81953857338575872	28452708250169344	5995206851792896	795335666925568	67433696788480	3627364122624	119184293888	2172649472	16777216			413920896501672000
13	778700428129000000	6219904229560320000	16700856199564000000	19963635654920576000	12421451089896524800	4453458125628858368	983115826510647296	139304499111378944	12959228702031872	795702584934400	31791179104256	791771348992	11123294208	67108864		61674213578749128000
14	132379072781930000000	1060498520737770400000	2864025170844121280000	3460621486134753920000	2191501227902091520000	806773685716492021760	184943523009325473792	27614228154977009664	2760286875790934016	187106354246975488	8587310787461120	261765845745664	5058045411328	55901683712	268435456	10731313162702348272000

Computed in your browser ?

Technical details

Node id: n-afff5b8d53e95899 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-7a12baa115b5eeac

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 ?

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:          1;
  1:          1,           4;
  2:          2,          12,           16;
  3:         10,          68,          128,           64;
  4:        100,         720,         1552,         1152,         256;
  5:       1700,       12640,        29264,        25792,        8960,        1024;
  6:      44200,      335440,       811424,       787648,      336128,       62464,       4096;
  7:    1635400,    12588080,     31364448,     32388672,    15587328,     3655680,     401408,     16384;
  8:   81770000,   635945600,   1618574720,   1744891392,   908921088,   245133312,   34693120,   2424832,    65536;
  9: 5315050000, 41663544000, 107751139200, 119892239360, 66059436288, 19569349632, 3235586048, 296386560, 13959168, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (n^2-2*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[(n^2 - 2*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] = (n^2 - 2*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-424ddc300423e142
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-424ddc300423e142
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-424ddc300423e142

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-424ddc300423e142

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-aa0ec4b5da096d18 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.