Real-rooted atlas

A266213: Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.

A266213 nonnegative coefficients

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

Recurrence from OEIS A266213
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2123
31427
4168217
5181812241
6110323816299
7112508866202239

Show all 15 stored rows

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

Computed in your browser ?

Technical details
Node id
n-a1540a6d32c29f8f · revision 1 · JSON
Definition source
oeis_catalog:A266213
Fingerprint
f6-f7ea49607396af2f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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: 1;
  2: 1,  2;
  3: 1,  4,  2;
  4: 1,  6,  8,   2;
  5: 1,  8, 18,  12,   2;
  6: 1, 10, 32,  38,  16,   2;
  7: 1, 12, 50,  88,  66,  20,   2;
  8: 1, 14, 72, 170, 192, 102,  24,  2;
  9: 1, 16, 98, 292, 450, 360, 146, 28, 2;
%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.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
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; 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.

A035607: 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
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 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*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 + 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.

A113413: reversed 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
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 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*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 + 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 >= 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-1e993b9b3c7faf34
  • 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-1e993b9b3c7faf34
  • 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-1e993b9b3c7faf34
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-1e993b9b3c7faf34
  • 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-b76eec14ee8e8c71
  • 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-b76eec14ee8e8c71

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 12 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