A269944 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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	1														2
2	2	4	1													7
3	10	26	11	1												48
4	100	296	158	24	1											579
5	1700	5428	3298	638	45	1										11110
6	44200	148256	97772	21800	1988	76	1									314093
7	1635400	5677928	3961364	969772	103308	5180	119	1								12353072
8	81770000	291209728	211668856	55359280	6548404	388208	11844	176	1							646956497
9	5315050000	19301612048	14473023080	3976099896	507199156	33722964	1229132	24516	249	1						43607961042
10	435834100000	1607348849984	1235035550768	352441514240	47595227312	3441097024	141886580	3411056	46926	340	1					3681841684231
11	44019244100000	164385416798368	128816010579088	37888953031728	5349940382000	412351511856	18622961084	510280628	8525990	84326	451	1				380891058255520
12	5370347780200000	20263425510299264	16137570728605280	4864935139545088	711981441163728	57718582387712	2796090530608	84449162096	1618659328	19572696	143858	584	1			47408861360270243
13	778700428129000000	2963830472283892544	2392486322615275424	736147971381278304	110950169872940560	9369768799320528	479928252509520	15632363169200	332104039280	4632854512	41870686	234962	741	1		6991981030476386262
14	132379072781930000000	507593711188674625024	414471477962111265472	129746085371576422016	20041477529272935744	1750659709754032960	93837141240996048	3246856533457920	74746882161040	1161384996928	12169577992	84398808	369824	924	1	1206079644461615240701

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Technical details

Node id: n-abc0ba18569a86e9 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-9edb451024733e5b

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:          1,           1;
  2:          2,           4,           1;
  3:         10,          26,          11,          1;
  4:        100,         296,         158,         24,         1;
  5:       1700,        5428,        3298,        638,        45,        1;
  6:      44200,      148256,       97772,      21800,      1988,       76,       1;
  7:    1635400,     5677928,     3961364,     969772,    103308,     5180,     119,     1;
  8:   81770000,   291209728,   211668856,   55359280,   6548404,   388208,   11844,   176,   1;
  9: 5315050000, 19301612048, 14473023080, 3976099896, 507199156, 33722964, 1229132, 24516, 249, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n^2-2*n+2+x)*P(n-1,x) + x*d/dx 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 + x)*P[n - 1, x] + x*D[P[n - 1, x], 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 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-0687c4d1ba7aff91
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-0687c4d1ba7aff91
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-0687c4d1ba7aff91

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 A008277 e-0687c4d1ba7aff91

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- 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.

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741