A103633 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 4\,x + x^{2}\right)\,P_{n-2}(x) + \left(4\,x + 2\,x^{2}\right)\,P_{n-2}'(x) + \left(x^{2}\right)\,P_{n-2}''(x)\)

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

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

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	2	10	7	1												20
4	4	32	38	12	1											87
5	4	68	146	86	17	1										322
6	8	208	652	576	188	24	1									1657
7	8	424	2164	2956	1516	332	31	1								7432
8	16	1280	9080	16944	12052	3840	580	40	1							43833
9	16	2576	28520	76856	77204	35092	7900	900	49	1						229114
10	32	7744	116656	412800	540080	322848	98292	16000	1390	60	1					1515903
11	32	15520	357712	1767856	3113200	2477168	1010892	226292	28510	1990	71	1				8999244
12	64	46592	1446368	9196992	20447056	20453376	10564304	3047520	511392	50400	2842	84	1			65766991
13	64	93248	4385696	38234336	111432272	143167312	94403504	34944464	7650048	1015392	81662	3850	97	1		435411946
14	128	279808	17636032	195557376	706827968	1113603456	898395344	408903680	111444944	18819360	1995336	131712	5208	112	1	3473600465

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Technical details

Node id: n-7b0e235a9b3149b1 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-a8e03434bb408ebe

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:  2,   10,     7,     1;
  4:  4,   32,    38,    12,     1;
  5:  4,   68,   146,    86,    17,     1;
  6:  8,  208,   652,   576,   188,    24,    1;
  7:  8,  424,  2164,  2956,  1516,   332,   31,   1;
  8: 16, 1280,  9080, 16944, 12052,  3840,  580,  40,  1;
  9: 16, 2576, 28520, 76856, 77204, 35092, 7900, 900, 49, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+4*x+x^2)*P(n-2,x) + (4*x+2*x^2)*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x + x^2)*P[n - 2, x] + (4*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 4*x + x^2)*v[i - 2] + (4*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), 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-d2f735ac3154f341
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-d2f735ac3154f341
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-d2f735ac3154f341

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-d2f735ac3154f341

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-35a33ce2a6b89415 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