Real-rooted atlas

A122848: Exponential Riordan array (1, x(1+x/2)).

A100861 → reverse

A122848 nonnegative coefficients

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

Recurrence from OEIS A122848
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(-x + n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20112
300314
40036110
50001510126
6000154515176
70000105105211232

Show all 15 stored rows

Generated b-files: A000085 (row sums), A100861 (reversed triangle), A122848, A144299 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-f6268da8d9636534 · revision 1 · JSON
Definition source
oeis_catalog:A122848
Fingerprint
f6-eccca8d515312e3b

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 ?

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: 0, 1;
  2: 0, 1, 1;
  3: 0, 0, 3,  1;
  4: 0, 0, 3,  6,   1;
  5: 0, 0, 0, 15,  10,   1;
  6: 0, 0, 0, 15,  45,  15,    1;
  7: 0, 0, 0,  0, 105, 105,   21,   1;
  8: 0, 0, 0,  0, 105, 420,  210,  28,  1;
  9: 0, 0, 0,  0,   0, 945, 1260, 378, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
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] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n - 1)*x*v[i - 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.

A100861: reversed triangle, offset 0, from the row P0, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1,  1;
  3: 1,  3;
  4: 1,  6,   3;
  5: 1, 10,  15;
  6: 1, 15,  45,   15;
  7: 1, 21, 105,  105;
  8: 1, 28, 210,  420, 105;
  9: 1, 36, 378, 1260, 945;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n - 1)*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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.

A122848: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 1;
  3: 0, 0, 3,  1;
  4: 0, 0, 3,  6,   1;
  5: 0, 0, 0, 15,  10,   1;
  6: 0, 0, 0, 15,  45,  15,    1;
  7: 0, 0, 0,  0, 105, 105,   21,   1;
  8: 0, 0, 0,  0, 105, 420,  210,  28,  1;
  9: 0, 0, 0,  0,   0, 945, 1260, 378, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
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] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n - 1)*x*v[i - 2]); 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.

A144299: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  0;
  2: 1,  1,   0;
  3: 1,  3,   0,    0;
  4: 1,  6,   3,    0,   0;
  5: 1, 10,  15,    0,   0, 0;
  6: 1, 15,  45,   15,   0, 0, 0;
  7: 1, 21, 105,  105,   0, 0, 0, 0;
  8: 1, 28, 210,  420, 105, 0, 0, 0, 0;
  9: 1, 36, 378, 1260, 945, 0, 0, 0, 0, 0;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = x*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
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] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (n - 1)*x*v[i - 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-01d50252f4aaa594
  • 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-01d50252f4aaa594
  • 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-01d50252f4aaa594

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

Used in 12 proof steps

Same fingerprint (2)

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

References

  1. [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