basis-map coefficients of A021012 → basis A013611

only rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-1	-4							-5
3	2	16	32						50
4	-6	-72	-288	-384					-750
5	24	384	2304	6144	6144				15000
6	-120	-2400	-19200	-76800	-153600	-122880			-375000
7	720	17280	172800	921600	2764800	4423680	2949120		11250000
8	-5040	-141120	-1693440	-11289600	-45158400	-108380160	-144506880	-82575360	-393750000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-6e0864853f49fb26 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-03d7b8417365c509

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 -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       1;
   2:      -1,        -4;
   3:       2,        16,         32;
   4:      -6,       -72,       -288,        -384;
   5:      24,       384,       2304,        6144,         6144;
   6:    -120,     -2400,     -19200,      -76800,      -153600,      -122880;
   7:     720,     17280,     172800,      921600,      2764800,      4423680,       2949120;
   8:   -5040,   -141120,   -1693440,   -11289600,    -45158400,   -108380160,    -144506880,     -82575360;
   9:   40320,   1290240,   18063360,   144506880,    722534400,   2312110080,    4624220160,    5284823040,    2642411520;
  10: -362880, -13063680, -209018880, -1950842880, -11705057280, -46820229120, -124853944320, -214035333120, -214035333120, -95126814720;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n - 1 + (4*n - 4)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(n - 1 + (4*n - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-e262ea1acc16711f
Coefficients of one sign. Every nonzero row, n >= 1: 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-e262ea1acc16711f
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-e262ea1acc16711f

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-e262ea1acc16711f

Geometric basis map (elementary) polytool ?
- premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-51f2fe49bf45c5d7 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image mobius, shift 0 e-bb5cb95fed86aa9c
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers root_domain e-51f2fe49bf45c5d7 (expanded above)

Related sequences ?

Same fingerprint (14)

Equal up to content, sign, reversal and a small index shift.