A249252 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-4 + 4\,x + 2\,x^{2} + 4\,n - 8\,n x - 2\,n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x + 30\,x^{2} + 4\,x^{3} - 4\,n x - 12\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(4\,x^{2} + 12\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 4 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	1														5
2	8	8	1													17
3	16	32	12	1												61
4	32	96	72	16	1											217
5	64	256	304	128	20	1										773
6	128	640	1056	704	200	24	1									2753
7	256	1536	3264	3072	1360	288	28	1								9805
8	512	3584	9344	11520	7200	2336	392	32	1							34921
9	1024	8192	25344	38912	32064	14592	3696	512	36	1						124373
10	2048	18432	66048	121856	126080	75648	26656	5504	648	40	1					442961
11	4096	40960	166912	360448	451840	341504	158144	45056	7824	800	44	1				1577629
12	8192	90112	411648	1019904	1507840	1387008	809088	301568	71712	10720	968	48	1			5618809
13	16384	196608	995328	2785280	4756480	5185536	3688192	1728512	535104	108800	14256	1152	52	1		20011685
14	32768	425984	2367488	7389184	14338048	18143232	15335936	8763392	3401856	896128	158752	18496	1352	56	1	71272673

Generated b-files: A007483 (row sums), A249252 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-95035df316bee7cb · revision 1 · JSON
Definition source: derived:reversal
Fingerprint: f6-a9505c52a70b77f5

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 ?

A007483: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A249252: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

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:    4,    1;
  2:    8,    8,     1;
  3:   16,   32,    12,     1;
  4:   32,   96,    72,    16,     1;
  5:   64,  256,   304,   128,    20,     1;
  6:  128,  640,  1056,   704,   200,    24,    1;
  7:  256, 1536,  3264,  3072,  1360,   288,   28,   1;
  8:  512, 3584,  9344, 11520,  7200,  2336,  392,  32,  1;
  9: 1024, 8192, 25344, 38912, 32064, 14592, 3696, 512, 36, 1;

%F Formula

The row polynomials satisfy (2*n-2)*P(n,x) = (4*n-4+(4*n^2-8*n+4)*x-(2*n-2)*x^2)*P(n-1,x) - ((4*n-4)*x+(12*n-30)*x^2+(n-4)*x^3)*d/dx P(n-1,x) + (4*x^2+12*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (4*n^2 - 8*n + 4)*x - (2*n - 2)*x^2)*P[n - 1, x] - ((4*n - 4)*x + (12*n - 30)*x^2 + (n - 4)*x^3)*D[P[n - 1, x], x] + (4*x^2 + 12*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(2*n - 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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (4*n^2 - 8*n + 4)*x - (2*n - 2)*x^2)*v[i - 1] - ((4*n - 4)*x + (12*n - 30)*x^2 + (n - 4)*x^3)*deriv(v[i - 1], x) + (4*x^2 + 12*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(2*n - 2)); 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.

A249252: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  8,   8;
  3: 1, 12,  32,   16;
  4: 1, 16,  72,   96,    32;
  5: 1, 20, 128,  304,   256,    64;
  6: 1, 24, 200,  704,  1056,   640,   128;
  7: 1, 28, 288, 1360,  3072,  3264,  1536,   256;
  8: 1, 32, 392, 2336,  7200, 11520,  9344,  3584,  512;
  9: 1, 36, 512, 3696, 14592, 32064, 38912, 25344, 8192, 1024;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-2)*P(n,x) = (4*n-4+(4*n^2-8*n+4)*x-(2*n-2)*x^2)*P(n-1,x) - ((4*n-4)*x+(12*n-30)*x^2+(n-4)*x^3)*d/dx P(n-1,x) + (4*x^2+12*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (4*n^2 - 8*n + 4)*x - (2*n - 2)*x^2)*P[n - 1, x] - ((4*n - 4)*x + (12*n - 30)*x^2 + (n - 4)*x^3)*D[P[n - 1, x], x] + (4*x^2 + 12*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (4*n^2 - 8*n + 4)*x - (2*n - 2)*x^2)*v[i - 1] - ((4*n - 4)*x + (12*n - 30)*x^2 + (n - 4)*x^3)*deriv(v[i - 1], x) + (4*x^2 + 12*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(2*n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-8565488d6e1a62c1
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-8565488d6e1a62c1
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-8565488d6e1a62c1

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

preserver_image reversal, shift 0 e-8565488d6e1a62c1

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-fb471901eb718e65 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A249252: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003