Real-rooted atlas

A122542: Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

A122542 nonnegative coefficients

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

Recurrence from OEIS A122542
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30213
402417
50286117
60212188141
70216383210199
80220668850121239

Show all 15 stored rows

Generated b-files: A001333 (row sums), A035607 (reversed triangle), A078057 (row sums), A113413, A122542, A266213 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-b0fc620949fb5990 · revision 1 · JSON
Definition source
oeis_catalog:A122542
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 ?

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: 0, 1;
   3: 0, 2,  1;
   4: 0, 2,  4,   1;
   5: 0, 2,  8,   6,   1;
   6: 0, 2, 12,  18,   8,   1;
   7: 0, 2, 16,  38,  32,  10,   1;
   8: 0, 2, 20,  66,  88,  50,  12,  1;
   9: 0, 2, 24, 102, 192, 170,  72, 14,  1;
  10: 0, 2, 28, 146, 360, 450, 292, 98, 16, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = (1 + x)*v[i - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

A035607: reversed triangle, offset 0, from the row P2, low zeros dropped
%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, down to the lowest nonzero coefficient, where P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 2*x + x^2.
%t Mathematica
P[0, x_] := x;
P[1, x_] := 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 2*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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 P2, low zeros dropped
%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
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 2*x + x^2.
%t Mathematica
P[0, x_] := x;
P[1, x_] := 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 2*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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.

A122542: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 2,  1;
  3: 0, 2,  4,   1;
  4: 0, 2,  8,   6,   1;
  5: 0, 2, 12,  18,   8,   1;
  6: 0, 2, 16,  38,  32,  10,   1;
  7: 0, 2, 20,  66,  88,  50,  12,  1;
  8: 0, 2, 24, 102, 192, 170,  72, 14,  1;
  9: 0, 2, 28, 146, 360, 450, 292, 98, 16, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 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[(1 + x)*P[n - 1, x] + 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] = (1 + x)*v[i - 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.

A266213: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,  0;
  2: 1,  2,  0;
  3: 1,  4,  2,   0;
  4: 1,  6,  8,   2,   0;
  5: 1,  8, 18,  12,   2,   0;
  6: 1, 10, 32,  38,  16,   2,   0;
  7: 1, 12, 50,  88,  66,  20,   2,  0;
  8: 1, 14, 72, 170, 192, 102,  24,  2, 0;
  9: 1, 16, 98, 292, 450, 360, 146, 28, 2, 0;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 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[(1 + x)*P[n - 1, x] + 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] = (1 + x)*v[i - 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 >= 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-e90fe0b3eba80839
  • 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-e90fe0b3eba80839
  • 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-e90fe0b3eba80839
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-e90fe0b3eba80839
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(2), -1 + sqrt(2)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has a real-rooted image
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-28dfe6801aeae3ea
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(2), 0] to roots in (-inf, 0]: every PF input with all roots in [-1 - sqrt(2), 0] has a PF image; no larger [-R, 0] works
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-28dfe6801aeae3ea

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 14 proof steps

Same fingerprint (6)

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