A038281 → 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

\(P_{n}(x) = \left(11 + 9\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 121 + 198\,x + 81\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	121	198	81								400
1	1331	3267	2673	729							8000
2	14641	47916	58806	32076	6561						160000
3	161051	658845	1078110	882090	360855	59049					3200000
4	1771561	8696754	17788815	19405980	11908215	3897234	531441				64000000
5	19487171	111608343	273947751	373565115	305644185	150043509	40920957	4782969			1280000000
6	214358881	1403076312	4017900348	6574746024	6724172070	4401276264	1800522108	420901272	43046721		25600000000
7	2357947691	17363069361	56824590636	108483309396	133138606986	108931587534	59417229564	20834612964	4261625379	387420489	512000000000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-4113d906b9d95b00 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-6bae7ffb6b8d3725

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

A009964: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:          121,           198,             81;
  1:         1331,          3267,           2673,            729;
  2:        14641,         47916,          58806,          32076,           6561;
  3:       161051,        658845,        1078110,         882090,         360855,          59049;
  4:      1771561,       8696754,       17788815,       19405980,       11908215,        3897234,         531441;
  5:     19487171,     111608343,      273947751,      373565115,      305644185,      150043509,       40920957,        4782969;
  6:    214358881,    1403076312,     4017900348,     6574746024,     6724172070,     4401276264,     1800522108,      420901272,      43046721;
  7:   2357947691,   17363069361,    56824590636,   108483309396,   133138606986,   108931587534,    59417229564,    20834612964,    4261625379,     387420489;
  8:  25937424601,  212215292190,   781338121245,  1704737719080,  2440874461410,  2396494925748,  1633973813010,   763935808680,  234389395845,   42616253790,   3486784401;
  9: 285311670611, 2567805035499, 10504656963405, 25784158001085, 42192258547230, 48329314335918, 39542166274842, 23109058212570, 9453705632415, 2578283354295, 421900912521, 31381059609;

%F Formula

The row polynomials satisfy P(n,x) = (11+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 198*x + 81*x^2.

%t Mathematica

P[0, x_] := 121 + 198*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 9*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] = 121 + 198*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 9*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-f67f09ded1c0d580
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-f67f09ded1c0d580
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-f67f09ded1c0d580

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-f67f09ded1c0d580

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-da334b7c58401d78 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-c5656743efdffc45 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003