A272867 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 12\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	12	4													22
2	36	144	192	96	16											484
3	216	1296	3024	3456	2016	576	64									10648
4	1296	10368	34560	62208	65664	41472	15360	3072	256							234256
5	7776	77760	336960	829440	1278720	1285632	852480	368640	99840	15360	1024					5153632
6	46656	559872	2985984	9331200	18973440	26376192	25657344	17584128	8432640	2764800	589824	73728	4096			113379904
7	279936	3919104	24820992	94058496	237758976	423263232	546352128	518897664	364234752	188116992	70447104	18579456	3268608	344064	16384	2494357888

Show all 15 stored rows

Generated b-files: A009966 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-2422ccdf8fc66e9a · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-3c4a6b4bc3495597

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

OEIS matches ?

A009966: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:        6,        12,          4;
  2:       36,       144,        192,         96,          16;
  3:      216,      1296,       3024,       3456,        2016,         576,           64;
  4:     1296,     10368,      34560,      62208,       65664,       41472,        15360,         3072,          256;
  5:     7776,     77760,     336960,     829440,     1278720,     1285632,       852480,       368640,        99840,        15360,         1024;
  6:    46656,    559872,    2985984,    9331200,    18973440,    26376192,     25657344,     17584128,      8432640,      2764800,       589824,       73728,        4096;
  7:   279936,   3919104,   24820992,   94058496,   237758976,   423263232,    546352128,    518897664,    364234752,    188116992,     70447104,    18579456,     3268608,      344064,      16384;
  8:  1679616,  26873856,  197074944,  877879296,  2654539776,  5768921088,   9308307456,  11362664448,  10597588992,   7575109632,   4137025536,  1709309952,   524353536,   115605504,   17301504,   1572864,    65536;
  9: 10077696, 181398528, 1511654400, 7739670528, 27250089984, 69979521024, 135695056896, 202951360512, 237170737152, 218072383488, 158113824768, 90200604672, 40205942784, 13823115264, 3588489216, 679477248, 88473600, 7077888, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (6+12*x+4*x^2)*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[(6 + 12*x + 4*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; for(n = 1, N, my(i = n + 1); v[i] = (6 + 12*x + 4*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-e79851ce4a3e511f
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-e79851ce4a3e511f
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-e79851ce4a3e511f

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 A013609 e-e79851ce4a3e511f

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-eab26daf7024ea74 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

A272867 → basis A013610