A299989 → 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

\(P_{n}(x) = \left(8 + 18\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	1	3															4
1	8	42	63	27													140
2	64	480	1332	1728	1053	243											4900
3	512	4992	19872	42120	51516	36450	13851	2187									171500
4	4096	49152	253440	739584	1349136	1597968	1230552	594864	164025	19683							6002500
5	32768	466944	2949120	10920960	26386560	43724448	50750064	41290560	23094720	8463690	1830519	177147					210087500
6	262144	4325376	32292864	144654336	434211840	923042304	1430519616	1637345664	1384738416	855029520	374843052	110539728	19663317	1594323			7353062500
7	2097152	39321600	338558976	1777434624	6368108544	16502040576	31966824960	47155499136	53424805824	46501638624	30851921520	15326758440	5520609108	1361551842	205667667	14348907	257357187500

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-8624395344cd3600 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-05ba62c68bfd52e0

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
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,          3;
  1:         8,         42,          63,           27;
  2:        64,        480,        1332,         1728,          1053,           243;
  3:       512,       4992,       19872,        42120,         51516,         36450,          13851,           2187;
  4:      4096,      49152,      253440,       739584,       1349136,       1597968,        1230552,         594864,         164025,          19683;
  5:     32768,     466944,     2949120,     10920960,      26386560,      43724448,       50750064,       41290560,       23094720,        8463690,        1830519,         177147;
  6:    262144,    4325376,    32292864,    144654336,     434211840,     923042304,     1430519616,     1637345664,     1384738416,      855029520,      374843052,      110539728,       19663317,        1594323;
  7:   2097152,   39321600,   338558976,   1777434624,    6368108544,   16502040576,    31966824960,    47155499136,    53424805824,    46501638624,    30851921520,    15326758440,     5520609108,     1361551842,      205667667,      14348907;
  8:  16777216,  352321536,  3435134976,  20667432960,   85985722368,  262639190016,   610084306944,  1101165207552,  1563898855680,  1758059106048,  1564668119808,  1096463402496,   597713818464,   248204204640,    75838756464,   16070775840,   2109289329,    129140163;
  9: 134217728, 3120562176, 33973862400, 230342787072, 1090815787008, 3834863419392, 10382051377152, 22154591895552, 37822923343872, 52125139118592, 58237498568448, 52758265330944, 38600064870912, 22612652991936, 10453810101408, 3727501664832, 988697087928, 183637311786, 21308126895, 1162261467;

%F Formula

The row polynomials satisfy P(n,x) = (8+18*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(8 + 18*x + 9*x^2)*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 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 18*x + 9*x^2)*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-78fc61375bb1ec88
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-78fc61375bb1ec88
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-78fc61375bb1ec88

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-78fc61375bb1ec88

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-0e7ef206328a472c root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.