Real-rooted atlas

A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.

A391533 nonnegative coefficients only rational roots

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 A391533
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 4 + 4\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04419
1485118
2412136136
341625197172
442041442681144
54246185703491288
64288514615510443101576
7432113231301259147531111152

Show all 15 stored rows

Generated b-files: A005010 (row sums), A391533 ?

Computed in your browser ?

Technical details
Node id
n-f48bed27dbc29c5f · revision 1 · JSON
Definition source
oeis_catalog:A391533
Fingerprint
f6-a681886699e85b8d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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): yes

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: 4,  4,   1;
  1: 4,  8,   5,   1;
  2: 4, 12,  13,   6,   1;
  3: 4, 16,  25,  19,   7,    1;
  4: 4, 20,  41,  44,  26,    8,   1;
  5: 4, 24,  61,  85,  70,   34,   9,   1;
  6: 4, 28,  85, 146, 155,  104,  43,  10,   1;
  7: 4, 32, 113, 231, 301,  259, 147,  53,  11,  1;
  8: 4, 36, 145, 344, 532,  560, 406, 200,  64, 12,  1;
  9: 4, 40, 181, 489, 876, 1092, 966, 606, 264, 76, 13, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 4*x + x^2.
%t Mathematica
P[0, x_] := 4 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4 + 4*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1]); 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.

A391533: triangle, offset 2, from the row P0
%e Example
Triangle begins:
   2: 4,  4,   1;
   3: 4,  8,   5,   1;
   4: 4, 12,  13,   6,   1;
   5: 4, 16,  25,  19,   7,    1;
   6: 4, 20,  41,  44,  26,    8,   1;
   7: 4, 24,  61,  85,  70,   34,   9,   1;
   8: 4, 28,  85, 146, 155,  104,  43,  10,   1;
   9: 4, 32, 113, 231, 301,  259, 147,  53,  11,  1;
  10: 4, 36, 145, 344, 532,  560, 406, 200,  64, 12,  1;
  11: 4, 40, 181, 489, 876, 1092, 966, 606, 264, 76, 13, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 3, with P(2,x) = 4 + 4*x + x^2.
%t Mathematica
P[2, x_] := 4 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 4 + 4*x + x^2; for(n = 3, N, my(i = n - 1); v[i] = (1 + x)*v[i - 1]); 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(11)

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-3cff49c1888b5e80
  • 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-3cff49c1888b5e80, e-5e5f630c1e2baa8a
  • 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-3cff49c1888b5e80, e-5e5f630c1e2baa8a
  • 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-3cff49c1888b5e80
  • 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-5e5f630c1e2baa8a
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
    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-408f6d5a7bdb6c51
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 1] to roots in (-inf, 0]: an input root r <= 1 gives the root (r - a)/d <= 0; a larger root gives a positive root
    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-408f6d5a7bdb6c51

Proof chains

3 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 16 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