A035607 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 + 8\,x + 4\,x^{2} + n - 8\,n x - 2\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 21\,x^{2} + 5\,x^{3} - n x - 6\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(x^{2} + 6\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)

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

\(P_{2}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	2	4	1						7
4	2	8	6	1					17
5	2	12	18	8	1				41
6	2	16	38	32	10	1			99
7	2	20	66	88	50	12	1		239
8	2	24	102	192	170	72	14	1	577

Show all 15 stored rows

Generated b-files: A035607 (reversed triangle), A078057 (row sums), A113413 ?

Computed in your browser ?

Technical details

Node id: n-63c9448cec96afa7 · revision 1 · JSON
Definition source: derived:reversal
Fingerprint: f6-f7ea49607396af2f

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 -1)
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 ?

A113413: triangle, 69 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A001333: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A035607: reversed triangle, 71 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A078057: 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 ?

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 2,  1;
   3: 2,  4,   1;
   4: 2,  8,   6,   1;
   5: 2, 12,  18,   8,    1;
   6: 2, 16,  38,  32,   10,   1;
   7: 2, 20,  66,  88,   50,  12,   1;
   8: 2, 24, 102, 192,  170,  72,  14,   1;
   9: 2, 28, 146, 360,  450, 292,  98,  16,  1;
  10: 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(2*n^2-8*n+8)*x-(2*n-4)*x^2)*P(n-1,x) - ((n-2)*x+(6*n-21)*x^2+(n-5)*x^3)*d/dx P(n-1,x) + (x^2+6*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (2*n^2 - 8*n + 8)*x - (2*n - 4)*x^2)*P[n - 1, x] - ((n - 2)*x + (6*n - 21)*x^2 + (n - 5)*x^3)*D[P[n - 1, x], x] + (x^2 + 6*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = ((n - 2 + (2*n^2 - 8*n + 8)*x - (2*n - 4)*x^2)*v[i - 1] - ((n - 2)*x + (6*n - 21)*x^2 + (n - 5)*x^3)*deriv(v[i - 1], x) + (x^2 + 6*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A035607: reversed triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  4,   2;
  3: 1,  6,   8,   2;
  4: 1,  8,  18,  12,   2;
  5: 1, 10,  32,  38,  16,    2;
  6: 1, 12,  50,  88,  66,   20,   2;
  7: 1, 14,  72, 170, 192,  102,  24,   2;
  8: 1, 16,  98, 292, 450,  360, 146,  28,  2;
  9: 1, 18, 128, 462, 912, 1002, 608, 198, 32, 2;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (n-1+(2*n^2-4*n+2)*x-(2*n-2)*x^2)*P(n-1,x) - ((n-1)*x+(6*n-15)*x^2+(n-4)*x^3)*d/dx P(n-1,x) + (x^2+6*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (2*n^2 - 4*n + 2)*x - (2*n - 2)*x^2)*P[n - 1, x] - ((n - 1)*x + (6*n - 15)*x^2 + (n - 4)*x^3)*D[P[n - 1, x], x] + (x^2 + 6*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (2*n^2 - 4*n + 2)*x - (2*n - 2)*x^2)*v[i - 1] - ((n - 1)*x + (6*n - 15)*x^2 + (n - 4)*x^3)*deriv(v[i - 1], x) + (x^2 + 6*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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.

A113413: triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0: 1;
  1: 2,  1;
  2: 2,  4,   1;
  3: 2,  8,   6,   1;
  4: 2, 12,  18,   8,    1;
  5: 2, 16,  38,  32,   10,   1;
  6: 2, 20,  66,  88,   50,  12,   1;
  7: 2, 24, 102, 192,  170,  72,  14,   1;
  8: 2, 28, 146, 360,  450, 292,  98,  16,  1;
  9: 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(2*n^2-4*n+2)*x-(2*n-2)*x^2)*P(n-1,x) - ((n-1)*x+(6*n-15)*x^2+(n-4)*x^3)*d/dx P(n-1,x) + (x^2+6*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (2*n^2 - 4*n + 2)*x - (2*n - 2)*x^2)*P[n - 1, x] - ((n - 1)*x + (6*n - 15)*x^2 + (n - 4)*x^3)*D[P[n - 1, x], x] + (x^2 + 6*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (2*n^2 - 4*n + 2)*x - (2*n - 2)*x^2)*v[i - 1] - ((n - 1)*x + (6*n - 15)*x^2 + (n - 4)*x^3)*deriv(v[i - 1], x) + (x^2 + 6*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #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 >= 1, 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-735e2173a8b41e53
Coefficients of one sign. Every nonzero row, n >= 1: 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-735e2173a8b41e53
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-735e2173a8b41e53

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-735e2173a8b41e53

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-f5199491b1877911 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

Same fingerprint (6)

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

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

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	2	4	1						7
4	2	8	6	1					17
5	2	12	18	8	1				41
6	2	16	38	32	10	1			99
7	2	20	66	88	50	12	1		239
8	2	24	102	192	170	72	14	1	577

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	2	4	1						7
4	2	8	6	1					17
5	2	12	18	8	1				41
6	2	16	38	32	10	1			99
7	2	20	66	88	50	12	1		239
8	2	24	102	192	170	72	14	1	577

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	2	4	1						7
4	2	8	6	1					17
5	2	12	18	8	1				41
6	2	16	38	32	10	1			99
7	2	20	66	88	50	12	1		239
8	2	24	102	192	170	72	14	1	577