A137320 → 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

\(P_{n}(x) = \left(1 + 8\,x + n\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	6	40	64													110
3	24	208	576	512												1320
4	120	1232	4544	7168	4096											17160
5	720	8352	37120	79360	81920	32768										240240
6	5040	64224	326656	852480	1208320	884736	262144									3603600
7	40320	554112	3127040	9433088	16486400	16744448	9175040	2097152								57657600
8	362880	5309568	32576256	109914112	223842304	282591232	216530944	92274688	16777216							980179200
9	3628800	55998720	368239104	1359751168	3117735936	4616650752	4426039296	2654994432	905969664	134217728						17643225600
10	39916800	645016320	4498619904	17903175680	45173104640	75725045760	85619638272	64613253120	31205621760	8724152320	1073741824					335221286400
11	479001600	8059530240	59143569408	250827067392	685302661120	1270085386240	1633236025344	1460316143616	891373486080	354334801920	82678120448	8589934592				6704425728000
12	6227020800	108605905920	833342644224	3733900431360	10915551133696	21993531310080	31392751419392	32049998069760	23270384467968	11737340313600	3909493981184	773094113280	68719476736			140792940288000
13	87178291200	1570298849280	12535644266496	58941347192832	182688919322624	395233847410688	615446770352128	699841984331776	582185367109632	350485840134144	148631638245376	42099269435392	7146825580544	549755813888		3097444686336000
14	1307674368000	24251909068800	200597054791680	984405362024448	3211864567382016	7390019065741312	12393572334567424	15421203927793664	14331516381298688	9914770538889216	5033361294753792	1820542147493888	443996539191296	65420941852672	4398046511104	71241227785728000

Generated b-files: A049398 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-07810086d0e6802d · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-027f711a8c8fc556

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 ?

A049398: 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:       2,        8;
  2:       6,       40,        64;
  3:      24,      208,       576,        512;
  4:     120,     1232,      4544,       7168,       4096;
  5:     720,     8352,     37120,      79360,      81920,      32768;
  6:    5040,    64224,    326656,     852480,    1208320,     884736,     262144;
  7:   40320,   554112,   3127040,    9433088,   16486400,   16744448,    9175040,    2097152;
  8:  362880,  5309568,  32576256,  109914112,  223842304,  282591232,  216530944,   92274688,  16777216;
  9: 3628800, 55998720, 368239104, 1359751168, 3117735936, 4616650752, 4426039296, 2654994432, 905969664, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+8*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[(n + 1 + 8*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] = (n + 1 + 8*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-40669049634a7e59
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-40669049634a7e59
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-40669049634a7e59

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-40669049634a7e59

Geometric basis map (elementary) polytool ?
- premise (sequence): Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-6aa1e3cc7e408461 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.