Real-rooted atlas

delannoy

palindromic γ-positive nonnegative coefficients

Proved for all n: all roots real, in [−3−2√2, −3+2√2] · coefficients of one sign · interlacing · Hadamard kernel basis map preserves real roots what this means

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21315
3155112
417137129
51925259170
6111416341111169
71136112912961131408

Show all 15 stored rows

Generated b-files: A008288, A008288 (reversed triangle), A069306 (row sums) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(x^{2} + n - n x - n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(9\,x^{2} + 3\,x^{3} - n x - 6\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(x^{2} + 6\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Image under the basis map A008288 · proved equal for all n (proof step)

\(P_{n}(x) = \left(x\right)\,P_{n-2}(x) + \left(1 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
Technical details
Node id
n-e00a1cb428776eaa · revision 3 · JSON
polytool family
delannoy
Definition source
named_family:delannoy
derived:reversal (equal by e-ce4f208b62242128)
derived:basis_map_A008288 (equal by e-9fd09d2b18d02ccc)
Fingerprint
f6-3c3dd961e8028a08

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: yes
  • γ-positive (gamma_positive): yes
  • 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: 1,  1;
  2: 1,  3,   1;
  3: 1,  5,   5,   1;
  4: 1,  7,  13,   7,   1;
  5: 1,  9,  25,  25,   9,   1;
  6: 1, 11,  41,  63,  41,  11,   1;
  7: 1, 13,  61, 129, 129,  61,  13,   1;
  8: 1, 15,  85, 231, 321, 231,  85,  15,  1;
  9: 1, 17, 113, 377, 681, 681, 377, 113, 17, 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) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 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, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 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};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A008288: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   1;
  3: 1,  5,   5,   1;
  4: 1,  7,  13,   7,   1;
  5: 1,  9,  25,  25,   9,   1;
  6: 1, 11,  41,  63,  41,  11,   1;
  7: 1, 13,  61, 129, 129,  61,  13,   1;
  8: 1, 15,  85, 231, 321, 231,  85,  15,  1;
  9: 1, 17, 113, 377, 681, 681, 377, 113, 17, 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) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 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] = 1 + 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.

A008288: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   1;
  3: 1,  5,   5,   1;
  4: 1,  7,  13,   7,   1;
  5: 1,  9,  25,  25,   9,   1;
  6: 1, 11,  41,  63,  41,  11,   1;
  7: 1, 13,  61, 129, 129,  61,  13,   1;
  8: 1, 15,  85, 231, 321, 231,  85,  15,  1;
  9: 1, 17, 113, 377, 681, 681, 377, 113, 17, 1;
%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) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 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] = 1 + 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 >= 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-15c233e4385971d1
  • 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-15c233e4385971d1, e-266d85ca03a298fa, e-33dba7e79fe9caa8, e-e771e35dc9ee3fbd
  • 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-15c233e4385971d1, e-266d85ca03a298fa, e-33dba7e79fe9caa8, e-e771e35dc9ee3fbd
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-15c233e4385971d1
  • 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-13cf2791da7ae36b
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(2), -1 + sqrt(2)] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]
    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-13cf2791da7ae36b
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(2), 0] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: 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-13cf2791da7ae36b
  • 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-266d85ca03a298fa
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1564556493/268435456, -46056243/268435456]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-e771e35dc9ee3fbd
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-33dba7e79fe9caa8

Proof chains

7 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 15 proof steps

Same fingerprint (1)

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