A114687 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	4	6	1						11
3	8	24	12	1					45
4	16	80	80	20	1				197
5	32	240	400	200	30	1			903
6	64	672	1680	1400	420	42	1		4279
7	128	1792	6272	7840	3920	784	56	1	20793

Show all 15 stored rows

Generated b-files: A114656, A114687 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-1b0fc3e5a40ede1a · revision 1 · JSON
Definition source: derived:reversal
Fingerprint: f6-9e57d05d9d9a1ce7

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 ?

A114687: reversed triangle, 54 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A114656: triangle, 54 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A001003: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:   2,     1;
  2:   4,     6,     1;
  3:   8,    24,    12,      1;
  4:  16,    80,    80,     20,      1;
  5:  32,   240,   400,    200,     30,     1;
  6:  64,   672,  1680,   1400,    420,    42,     1;
  7: 128,  1792,  6272,   7840,   3920,   784,    56,    1;
  8: 256,  4608, 21504,  37632,  28224,  9408,  1344,   72,  1;
  9: 512, 11520, 69120, 161280, 169344, 84672, 20160, 2160, 90, 1;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (2*n+4+3*n*x)*P(n-1,x) + (4*x-2*x^2)*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[Together[((2*n + 4 + 3*n*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 4 + 3*n*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x))/(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.

A114656: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1:   1;
   2:   2,     1;
   3:   4,     6,     1;
   4:   8,    24,    12,      1;
   5:  16,    80,    80,     20,      1;
   6:  32,   240,   400,    200,     30,     1;
   7:  64,   672,  1680,   1400,    420,    42,     1;
   8: 128,  1792,  6272,   7840,   3920,   784,    56,    1;
   9: 256,  4608, 21504,  37632,  28224,  9408,  1344,   72,  1;
  10: 512, 11520, 69120, 161280, 169344, 84672, 20160, 2160, 90, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (3*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n + 2 + (3*n - 3)*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A114687: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,    4;
   4: 1, 12,   24,     8;
   5: 1, 20,   80,    80,    16;
   6: 1, 30,  200,   400,   240,     32;
   7: 1, 42,  420,  1400,  1680,    672,     64;
   8: 1, 56,  784,  3920,  7840,   6272,   1792,   128;
   9: 1, 72, 1344,  9408, 28224,  37632,  21504,  4608,   256;
  10: 1, 90, 2160, 20160, 84672, 169344, 161280, 69120, 11520, 512;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (3*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n + 2 + (3*n - 3)*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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-0ad5e2ffb4ffd4ba
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-0ad5e2ffb4ffd4ba
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-0ad5e2ffb4ffd4ba

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-0ad5e2ffb4ffd4ba

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-0f37534727a0e0cd root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (5)

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

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023