A152842 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 40\,x + 48\,x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 4 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	12														16
2	8	40	48													96
3	32	256	672	576												1536
4	64	640	2368	3840	2304											9216
5	256	3328	17152	43776	55296	27648										147456
6	512	7680	47616	156160	285696	276480	110592									884736
7	2048	36864	282624	1196032	3016704	4534272	3760128	1327104								14155776
8	4096	81920	712704	3522560	10817536	21135360	25657344	17694720	5308416							84934656
9	16384	376832	3833856	22642688	85540864	214351872	356253696	378667008	233570304	63700992						1358954496
10	32768	819200	9175040	60620800	261652480	770867200	1569914880	2182348800	1981808640	1061683200	254803968					8153726976
11	131072	3670016	46530560	352583680	1774059520	6223298560	15530065920	27568373760	34115420160	28028436480	13759414272	3057647616				130459631616
12	262144	7864320	107741184	891289600	4958453760	19542835200	55953326080	117257011200	178504335360	192518553600	139632574464	61152952320	12230590464			782757789696
13	1048576	34603008	525336576	4858052608	30529290240	137672785920	458327326720	1140467957760	2121101475840	2912126238720	2868752941056	1920202702848	782757789696	146767085568		12524124635136
14	2097152	73400320	1189085184	11817451520	80490790912	397462732800	1467345797120	4114245222400	8804074782720	14308658380800	17386010836992	15315417169920	9246326390784	3424565329920	587068342272	75144747810816

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Technical details

Node id: n-899fb13ecc891703 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-b19b142556626fc3

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 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): 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;
  1:     4,     12;
  2:     8,     40,      48;
  3:    32,    256,     672,      576;
  4:    64,    640,    2368,     3840,     2304;
  5:   256,   3328,   17152,    43776,    55296,     27648;
  6:   512,   7680,   47616,   156160,   285696,    276480,    110592;
  7:  2048,  36864,  282624,  1196032,  3016704,   4534272,   3760128,   1327104;
  8:  4096,  81920,  712704,  3522560, 10817536,  21135360,  25657344,  17694720,   5308416;
  9: 16384, 376832, 3833856, 22642688, 85540864, 214351872, 356253696, 378667008, 233570304, 63700992;

%F Formula

The row polynomials satisfy P(n,x) = (8+40*x+48*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[(8 + 40*x + 48*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 40*x + 48*x^2)*v[i - 2]); 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-541f0d63dcdb93da
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-541f0d63dcdb93da
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-541f0d63dcdb93da

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-541f0d63dcdb93da

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-596b986e3a00db44 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.

A064861 → x ↦ x/(x+1)