A013612 → 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

\(P_{n}(x) = \left(6 + 15\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 36 + 180\,x + 225\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	36	180	225								441
1	216	1620	4050	3375							9261
2	1296	12960	48600	81000	50625						194481
3	7776	97200	486000	1215000	1518750	759375					4084101
4	46656	699840	4374000	14580000	27337500	27337500	11390625				85766121
5	279936	4898880	36741600	153090000	382725000	574087500	478406250	170859375			1801088541
6	1679616	33592320	293932800	1469664000	4592700000	9185400000	11481750000	8201250000	2562890625		37822859361
7	10077696	226748160	2267481600	13226976000	49601160000	124002900000	206671500000	221433750000	138396093750	38443359375	794280046581

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-31eaa59be7a07b63 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-9629d1e9c04dbfbe

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 ?

A009965: 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:        36,        180,          225;
  1:       216,       1620,         4050,         3375;
  2:      1296,      12960,        48600,        81000,         50625;
  3:      7776,      97200,       486000,      1215000,       1518750,         759375;
  4:     46656,     699840,      4374000,     14580000,      27337500,       27337500,       11390625;
  5:    279936,    4898880,     36741600,    153090000,     382725000,      574087500,      478406250,      170859375;
  6:   1679616,   33592320,    293932800,   1469664000,    4592700000,     9185400000,    11481750000,     8201250000,     2562890625;
  7:  10077696,  226748160,   2267481600,  13226976000,   49601160000,   124002900000,   206671500000,   221433750000,   138396093750,    38443359375;
  8:  60466176, 1511654400,  17006112000, 113374080000,  496011600000,  1488034800000,  3100072500000,  4428675000000,  4151882812500,  2306601562500,   576650390625;
  9: 362797056, 9976919040, 124711488000, 935336160000, 4676680800000, 16368382800000, 40920957000000, 73073137500000, 91341421875000, 76117851562500, 38058925781250, 8649755859375;

%F Formula

The row polynomials satisfy P(n,x) = (6+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 180*x + 225*x^2.

%t Mathematica

P[0, x_] := 36 + 180*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 15*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] = 36 + 180*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 15*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-46de8f4df4fd1cb3
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-46de8f4df4fd1cb3
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-46de8f4df4fd1cb3

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-46de8f4df4fd1cb3

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle of coefficients in expansion of (1+5x)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-8bdb70660e92b5b3 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-41a50f5538a6b055 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (6)

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