A286718 → 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

\(P_{n}(x) = \left(-1 + 4\,x + 3\,n\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	10	28	16						54
3	80	264	240	64					648
4	880	3224	3696	1664	256				9720
5	12320	48656	64640	38080	10240	1024			174960
6	209440	876432	1293504	905920	326400	58368	4096		3674160
7	4188800	18366400	29375808	23292416	10151680	2472960	315392	16384	88179840

Show all 15 stored rows

Generated b-files: A034001 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-7c95902cf1293962 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-449d2d9631fac89c

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 ?

A034001: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
A052690: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:          2,           4;
  2:         10,          28,          16;
  3:         80,         264,         240,          64;
  4:        880,        3224,        3696,        1664,         256;
  5:      12320,       48656,       64640,       38080,       10240,       1024;
  6:     209440,      876432,     1293504,      905920,      326400,      58368,      4096;
  7:    4188800,    18366400,    29375808,    23292416,    10151680,    2472960,    315392,     16384;
  8:   96342400,   439182400,   749109184,   653228800,   326658304,   97484800,  17145856,   1638400,   65536;
  9: 2504902400, 11804112000, 21233568384, 19980385536, 11106031104, 3841238016, 835731456, 111181824, 8257536, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (3*n-1+4*x)*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[(3*n - 1 + 4*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; for(n = 1, N, my(i = n + 1); v[i] = (3*n - 1 + 4*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-4c9f13aa30e5695d
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-4c9f13aa30e5695d
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-4c9f13aa30e5695d

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-4c9f13aa30e5695d

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-1844ef33b5ef77b8 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-576aa1dc371fa216 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - basis_map A007318 e-fe032cf3f375eb62
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers root_domain e-aa6a1f197b91b3b6 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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