A132393 → 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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	2	12	16													30
3	6	44	96	64												210
4	24	200	560	640	256											1680
5	120	1096	3600	5440	3840	1024										15120
6	720	7056	25984	47040	44800	21504	4096									151200
7	5040	52272	210112	433216	501760	329728	114688	16384								1663200
8	40320	438336	1889984	4306176	5746944	4644864	2236416	589824	65536							19958400
9	362880	4106304	18763200	46315520	68947200	64791552	38707200	14254080	2949120	262144						259459200
10	3628800	42514560	204057216	538208000	874734080	923704320	646238208	297369600	86507520	14417920	1048576					3632428800
11	39916800	482175360	2414687616	6736516864	11774906880	13659683840	10803437568	5856018432	2141061120	504627200	69206016	4194304				54486432000
12	479001600	5945771520	30904952832	90496952832	168244950016	211015833600	184279986176	113485971456	49116807168	14619770880	2848980992	327155712	16777216			871782912000
13	6227020800	79211036160	425547472896	1300080198144	2549172161536	3416185636864	3239703154688	2212437573632	1092462379008	386524250112	95515836416	15648948224	1526726656	67108864		14820309504000
14	87178291200	1133862589440	6274508765184	19903312665600	40888731054080	58023287562240	59020586713088	43932938649600	24144223600640	9781189017600	2883318710272	601148620800	83969966080	7046430720	268435456	266765571072000

Generated b-files: A001720 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-126effb6ce513816 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-5004f1ab8d505384

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 ?

A001720: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 4 ?

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:      1,       4;
  2:      2,      12,       16;
  3:      6,      44,       96,       64;
  4:     24,     200,      560,      640,      256;
  5:    120,    1096,     3600,     5440,     3840,     1024;
  6:    720,    7056,    25984,    47040,    44800,    21504,     4096;
  7:   5040,   52272,   210112,   433216,   501760,   329728,   114688,    16384;
  8:  40320,  438336,  1889984,  4306176,  5746944,  4644864,  2236416,   589824,   65536;
  9: 362880, 4106304, 18763200, 46315520, 68947200, 64791552, 38707200, 14254080, 2949120, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (n+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[(n + 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] = (n + 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-0a0b77ef6a10db4c
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-0a0b77ef6a10db4c
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-0a0b77ef6a10db4c

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-0a0b77ef6a10db4c

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-edbf92ee2e446027 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.