A269944 → 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

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

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

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	3														4
2	2	9	9													20
3	10	51	72	27												160
4	100	540	873	486	81											2080
5	1700	9480	16461	10881	2835	243										41600
6	44200	251580	456426	332289	106353	14823	729									1206400
7	1635400	9441060	17642502	13663971	4931928	867510	71442	2187								48256000
8	81770000	476959200	910448280	736126056	287588313	58171284	6174630	323676	6561							2557568000
9	5315050000	31247658000	60610015800	50579538480	20901618513	4643898399	575864802	39562830	1397493	19683						173914624000
10	435834100000	2578253106000	5063764269600	4329352202760	1865671333506	443504524257	61152608961	4971746466	233282916	5806485	59049					14782743040000
11	44019244100000	261711066006000	519174950547600	452455865287560	201420861292386	50390970950475	7506927077832	685604219949	38476813914	1286303733	23383404	177147				1537405276160000
12	5370347780200000	32060807785032000	64124477164825200	56757140416725120	25930712673533772	6751961039835108	1067018016346929	106164496067274	6750983957355	272359497168	6711686487	91762146	531441			192175659520000000
13	778700428129000000	4664928172170240000	9394231612254750000	8422158791919618000	3930224758912572300	1056826488796691976	174973495489810029	18594905978795517	1297386162018297	59745078961425	1790273032119	33440570631	352345383	1594323		28441997608960000000
14	132379072781930000000	795373890553327800000	1611014158599818220000	1459949689463099310000	693404685390896145000	191451177372175352820	32915973699657780858	3686054502864667977	276340365479497041	14048821909497141	483581652344505	11055716103627	160220427003	1328071059	4782969	4920465586350080000000

Computed in your browser ?

Technical details

Node id: n-26a8a6fa96e268b6 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-9f9eedba0e3776d0

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,           3;
  2:          2,           9,           9;
  3:         10,          51,          72,          27;
  4:        100,         540,         873,         486,          81;
  5:       1700,        9480,       16461,       10881,        2835,        243;
  6:      44200,      251580,      456426,      332289,      106353,      14823,       729;
  7:    1635400,     9441060,    17642502,    13663971,     4931928,     867510,     71442,     2187;
  8:   81770000,   476959200,   910448280,   736126056,   287588313,   58171284,   6174630,   323676,    6561;
  9: 5315050000, 31247658000, 60610015800, 50579538480, 20901618513, 4643898399, 575864802, 39562830, 1397493, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (n^2-2*n+2+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^2 - 2*n + 2 + 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^2 - 2*n + 2 + 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-6064ce9102e0184f
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-6064ce9102e0184f
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-6064ce9102e0184f

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-6064ce9102e0184f

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.