A269944 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

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

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

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	2														3
2	2	6	4													12
3	10	34	32	8												84
4	100	360	388	144	16											1008
5	1700	6320	7316	3224	560	32										19152
6	44200	167720	202856	98456	21008	1952	64									536256
7	1635400	6294040	7841112	4048584	974208	114240	6272	128								20913984
8	81770000	317972800	404643680	218111424	56807568	7660416	542080	18944	256							1087527168
9	5315050000	20831772000	26937784800	14986529920	4128714768	611542176	50556032	2315520	54528	512						72864320256
10	435834100000	1718835404000	2250561897600	1282771023040	368527670816	58403887968	5368678976	290984704	9102336	151040	1024					6120602901504
11	44019244100000	174474044004000	230744422465600	134060997122240	39786836798496	6635848026400	659044352512	40126813056	1501305344	33459712	405504	2048				630422098854912
12	5370347780200000	21373871856688000	28499767628811200	16816930493844480	5122116083660992	889147132817792	93675107059264	6213559897856	263412878080	7084695552	116390912	1060864	4096			78172340258009088
13	778700428129000000	3109952114780160000	4175214049891000000	2495454456865072000	776340693118532800	139170566425901824	15361184789228864	1088316399307648	50621987117312	1554106611200	31046073344	386607104	2715648	8192		11491334017927335936
14	132379072781930000000	530249260368885200000	716006292711030320000	432577685766844240000	136968826743880720000	25211677678640375680	2889742547020710528	215736157460757888	10782370608558336	365442098138624	8386045690880	127815354368	1234874368	6823936	16384	1976509451083501780992

Generated b-files: A375043 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-3255c4dcfe77aa0b · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-7d88dc66b737aeee

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 ?

A375043: reversed triangle, 42 terms agree, 7 full rows row lengths from the atlas (tabf) OEIS nonn · view at the OEIS · generated b-file 989 terms from n = 1 ?

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,           2;
  2:          2,           6,           4;
  3:         10,          34,          32,           8;
  4:        100,         360,         388,         144,         16;
  5:       1700,        6320,        7316,        3224,        560,        32;
  6:      44200,      167720,      202856,       98456,      21008,      1952,       64;
  7:    1635400,     6294040,     7841112,     4048584,     974208,    114240,     6272,     128;
  8:   81770000,   317972800,   404643680,   218111424,   56807568,   7660416,   542080,   18944,   256;
  9: 5315050000, 20831772000, 26937784800, 14986529920, 4128714768, 611542176, 50556032, 2315520, 54528, 512;

%F Formula

The row polynomials satisfy P(n,x) = (n^2-2*n+2+2*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^2 - 2*n + 2 + 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; for(n = 1, N, my(i = n + 1); v[i] = (n^2 - 2*n + 2 + 2*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.

A375043: reversed triangle, offset 1, from the row P₁

%e Example

Triangle begins:
   1:    2,      1;
   2:    4,      6,       2;
   3:    8,     32,      34,        10;
   4:   16,    144,     388,       360,        100;
   5:   32,    560,    3224,      7316,       6320,        1700;
   6:   64,   1952,   21008,     98456,     202856,      167720,        44200;
   7:  128,   6272,  114240,    974208,    4048584,     7841112,      6294040,       1635400;
   8:  256,  18944,  542080,   7660416,   56807568,   218111424,    404643680,     317972800,      81770000;
   9:  512,  54528, 2315520,  50556032,  611542176,  4128714768,  14986529920,   26937784800,   20831772000,    5315050000;
  10: 1024, 151040, 9102336, 290984704, 5368678976, 58403887968, 368527670816, 1282771023040, 2250561897600, 1718835404000, 435834100000;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n^2-2*n+2+2*x)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 2*x.

%t Mathematica

P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(n^2 - 2*n + 2 + 2*x)*P[n - 1, x]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 2*x; for(n = 2, N, my(i = n); v[i] = (n^2 - 2*n + 2 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-327e7f94e2e44f03
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-327e7f94e2e44f03
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-327e7f94e2e44f03

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-327e7f94e2e44f03

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-aa0ec4b5da096d18 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.