A089231: Triangular array A066667 or A008297 unsigned and transposed.

A089231 nonnegative coefficients

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

Recurrence from OEIS A089231

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	6	6						13
3	1	12	36	24					73
4	1	20	120	240	120				501
5	1	30	300	1200	1800	720			4051
6	1	42	630	4200	12600	15120	5040		37633
7	1	56	1176	11760	58800	141120	141120	40320	394353

Show all 15 stored rows

Generated b-files: A089231, A105278 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-d55979c30b97975a · revision 1 · JSON
Definition source: oeis_catalog:A089231
Fingerprint: f6-8fea465cea2922a0

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 ?

A089231: triangle, 45 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 ?
A105278: reversed triangle, 47 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 ?
A000262: 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: 1,  2;
  2: 1,  6,    6;
  3: 1, 12,   36,    24;
  4: 1, 20,  120,   240,    120;
  5: 1, 30,  300,  1200,   1800,     720;
  6: 1, 42,  630,  4200,  12600,   15120,     5040;
  7: 1, 56, 1176, 11760,  58800,  141120,   141120,    40320;
  8: 1, 72, 2016, 28224, 211680,  846720,  1693440,  1451520,   362880;
  9: 1, 90, 3240, 60480, 635040, 3810240, 12700800, 21772800, 16329600, 3628800;

%F Formula

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

A089231: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,    6;
   4: 1, 12,   36,    24;
   5: 1, 20,  120,   240,    120;
   6: 1, 30,  300,  1200,   1800,     720;
   7: 1, 42,  630,  4200,  12600,   15120,     5040;
   8: 1, 56, 1176, 11760,  58800,  141120,   141120,    40320;
   9: 1, 72, 2016, 28224, 211680,  846720,  1693440,  1451520,   362880;
  10: 1, 90, 3240, 60480, 635040, 3810240, 12700800, 21772800, 16329600, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) - 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[(1 + (2*n - 2)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
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] = (1 + (2*n - 2)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); 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.

A105278: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       36,       12,        1;
   5:     120,      240,      120,       20,       1;
   6:     720,     1800,     1200,      300,      30,      1;
   7:    5040,    15120,    12600,     4200,     630,     42,     1;
   8:   40320,   141120,   141120,    58800,   11760,   1176,    56,    1;
   9:  362880,  1451520,  1693440,   846720,  211680,  28224,  2016,   72,  1;
  10: 3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(2*n-2)*x)*P(n-1,x) - 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[(1 + (2*n - 2)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
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] = (1 + (2*n - 2)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); 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-db00758dc4aca02d
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-db00758dc4aca02d
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-db00758dc4aca02d
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-db00758dc4aca02d

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

theorem_check liu_wang e-db00758dc4aca02d root: theorem check

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 3 proof steps

e-a32213ca02c6584d obreschkoff_combination → obreschkoff_combination of Triangular array A066667 or A008297 unsigned and transposed.
e-a32213ca02c6584d obreschkoff_combination → obreschkoff_combination of Triangular array A066667 or A008297 unsigned and transposed.
e-cee0a9f908e74de4 preserver_image (affine, shift 0) → Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.

Related sequences ?

Same fingerprint (1)

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

A130534 → basis A008277

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