A026374 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 15\,x + 9\,x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	5	15	9													29
3	10	45	63	27												145
4	25	150	315	270	81											841
5	50	375	1080	1485	972	243										4205
6	125	1125	4050	7425	7290	3645	729									24389
7	250	2625	11475	27000	36855	29160	12393	2187								121945
8	625	7500	38250	108000	184275	194400	123930	43740	6561							707281
9	1250	16875	99000	330750	692550	941625	831060	459270	144342	19683						3536405
10	3125	46875	309375	1181250	2885625	4708125	5194125	3827250	1804275	492075	59049					20511149
11	6250	103125	759375	3290625	9315000	18073125	24512625	23236875	15090300	6396975	1594323	177147				102555745
12	15625	281250	2278125	10968750	34931250	77456250	122563125	139421250	113177250	63969750	23914845	5314410	531441			594823321
13	31250	609375	5400000	28771875	102768750	259706250	477495000	646531875	644618250	467471250	239738940	82373355	17006112	1594323		2974116605
14	78125	1640625	15750000	91546875	359690625	1009968750	2089040625	3232659375	3760273125	3272298750	2097715725	961022475	297606960	55801305	4782969	17249876309

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Technical details

Node id: n-15c34e3bd51729df · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-6ebcce6f82f6c3c8

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): 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:    2,     3;
  2:    5,    15,     9;
  3:   10,    45,    63,     27;
  4:   25,   150,   315,    270,     81;
  5:   50,   375,  1080,   1485,    972,    243;
  6:  125,  1125,  4050,   7425,   7290,   3645,    729;
  7:  250,  2625, 11475,  27000,  36855,  29160,  12393,   2187;
  8:  625,  7500, 38250, 108000, 184275, 194400, 123930,  43740,   6561;
  9: 1250, 16875, 99000, 330750, 692550, 941625, 831060, 459270, 144342, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (5+15*x+9*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[(5 + 15*x + 9*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] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (5 + 15*x + 9*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-fd8eb8561f83d912
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-fd8eb8561f83d912
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-fd8eb8561f83d912

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-fd8eb8561f83d912

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-6e3ca670df3d0a74 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image reversal, shift 0 e-b2e3690a28625635
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-6e3ca670df3d0a74 (expanded above)

Related sequences ?

No candidate equalities or related sequences.