A137320 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	6	14	4													24
3	24	96	60	8												188
4	120	720	732	208	16											1796
5	720	6000	8760	3960	640	32										20112
6	5040	55440	108360	69000	17520	1824	64									257248
7	40320	564480	1411200	1182720	418320	67872	4928	128								3689968
8	362880	6289920	19474560	20563200	9476880	2126208	239232	12800	256							58545936
9	3628800	76204800	285223680	367960320	211710240	61477920	9515520	785664	32256	512						1016539712
10	39916800	997920000	4430764800	6825772800	4758415200	1714456800	341812800	38672640	2442240	79360	1024					19150254464
11	479001600	14050713600	72888076800	131725440000	108819849600	47234880000	11632420800	1689114240	145728000	7265280	191488	2048				388672683456
12	6227020800	211718707200	1267198732800	2648559513600	2548667721600	1304041939200	385280280000	68870926080	7604340480	516679680	20849664	454656	4096			8448707169856
13	87178291200	3399953356800	23233014604800	55505587737600	61367808902400	36394341984000	12625371158400	2698946490240	365872066560	31742430720	1742247936	58066944	1064960	8192		195711618410752
14	1307674368000	57973563648000	448227184204800	1212083371699200	1522470780230400	1032594167404800	413673705244800	103520190533760	16739927043840	1779244346880	124463539200	5632972800	157667328	2465792	16384	4810500065385984

Computed in your browser ?

Technical details

Node id: n-0d4d6515fe1b63b7 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-b8f53010a9224832

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:       2,        2;
  2:       6,       14,         4;
  3:      24,       96,        60,         8;
  4:     120,      720,       732,       208,        16;
  5:     720,     6000,      8760,      3960,       640,       32;
  6:    5040,    55440,    108360,     69000,     17520,     1824,      64;
  7:   40320,   564480,   1411200,   1182720,    418320,    67872,    4928,    128;
  8:  362880,  6289920,  19474560,  20563200,   9476880,  2126208,  239232,  12800,   256;
  9: 3628800, 76204800, 285223680, 367960320, 211710240, 61477920, 9515520, 785664, 32256, 512;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+2*x)*P(n-1,x) + 2*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 + 1 + 2*x)*P[n - 1, x] + 2*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 + 1 + 2*x)*v[i - 1] + 2*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-658de135d35c8882
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-658de135d35c8882
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-658de135d35c8882

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-658de135d35c8882

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

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