A059344 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	3	4							7
3	7	12							19
4	25	72	48						145
5	81	280	240						601
6	331	1500	2160	960					4951
7	1303	6804	11760	6720					26587
8	5937	37072	84000	80640	26880				234529
9	26785	187632	489888	564480	241920				1510705
10	133651	1068660	3336480	5040000	3628800	967680			14175271
11	669351	5892700	20639520	35925120	31046400	10644480			104817571
12	3609673	35283864	141063120	293610240	332640000	191600640	42577920		1040385457
13	19674097	208837512	919261200	2146510080	2802159360	1937295360	553512960		8587250569
14	113525595	1313920972	6421663248	17115658560	26718531840	24216192000	11623772160	2214051840	89737316215

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Technical details

Node id: n-6fd4325317ef314b · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-3a08f59788b05985

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no

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:     1;
  2:     3,      4;
  3:     7,     12;
  4:    25,     72,     48;
  5:    81,    280,    240;
  6:   331,   1500,   2160,    960;
  7:  1303,   6804,  11760,   6720;
  8:  5937,  37072,  84000,  80640,  26880;
  9: 26785, 187632, 489888, 564480, 241920;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 2 + (4*n - 4)*x)*P[n - 2, 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] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (2*n - 2 + (4*n - 4)*x)*v[i - 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-f5d1118e248686fe
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-f5d1118e248686fe
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-f5d1118e248686fe
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-f5d1118e248686fe

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-f5d1118e248686fe

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers paper_theorem e-94e95c2ea0369f05 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.