Real-rooted atlas

A081581: Pascal-(1,6,1) array.

A081581 palindromic γ-positive 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 A081581
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
218110
311515132
412278221124
5129190190291440
61363518483513611624
7143561233923395614315888

Show all 15 stored rows

Generated b-files: A081581, A081581 (reversed triangle) ?

Computed in your browser ?

Other recurrences

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

\(\left(n\right)\,P_{n}(x) = \left(x^{2} + n - 6\,n x - n x^{2} + 7\,n^{2} x\right)\,P_{n-1}(x) + \left(39\,x^{2} + 3\,x^{3} - n x - 26\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(x^{2} + 26\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-212fa067c32397af · revision 2 · JSON
Definition source
oeis_catalog:A081581
derived:reversal (equal by e-7eaf14561efc7b42)
Fingerprint
f6-cb5ce4ca5d9f4d91

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,  8,    1;
  3: 1, 15,   15,    1;
  4: 1, 22,   78,   22,     1;
  5: 1, 29,  190,  190,    29,     1;
  6: 1, 36,  351,  848,   351,    36,    1;
  7: 1, 43,  561, 2339,  2339,   561,   43,    1;
  8: 1, 50,  820, 5006,  9766,  5006,  820,   50,  1;
  9: 1, 57, 1128, 9192, 28806, 28806, 9192, 1128, 57, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 6*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] + 6*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] + 6*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.

A081581: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  8,    1;
  3: 1, 15,   15,    1;
  4: 1, 22,   78,   22,     1;
  5: 1, 29,  190,  190,    29,     1;
  6: 1, 36,  351,  848,   351,    36,    1;
  7: 1, 43,  561, 2339,  2339,   561,   43,    1;
  8: 1, 50,  820, 5006,  9766,  5006,  820,   50,  1;
  9: 1, 57, 1128, 9192, 28806, 28806, 9192, 1128, 57, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 6*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] + 6*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] + 6*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.

A081581: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  8,    1;
  3: 1, 15,   15,    1;
  4: 1, 22,   78,   22,     1;
  5: 1, 29,  190,  190,    29,     1;
  6: 1, 36,  351,  848,   351,    36,    1;
  7: 1, 43,  561, 2339,  2339,   561,   43,    1;
  8: 1, 50,  820, 5006,  9766,  5006,  820,   50,  1;
  9: 1, 57, 1128, 9192, 28806, 28806, 9192, 1128, 57, 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) + 6*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] + 6*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] + 6*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-dfce858c9d52dee3
  • 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-5c5351a0cb2ba1df, e-dfce858c9d52dee3
  • 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-5c5351a0cb2ba1df, e-dfce858c9d52dee3
  • 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-dfce858c9d52dee3
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-6 - sqrt(42), -6 + sqrt(42)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-6 - sqrt(42), -6 + sqrt(42)] 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-d4a7a90d58b7b7be
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-6 - sqrt(42), -6 + sqrt(42)] to roots in [-13 - 2 sqrt(42), -13 + 2 sqrt(42)]: every real polynomial with all roots in E = [-6 - sqrt(42), -6 + sqrt(42)] has an image with all roots in I = [-13 - 2 sqrt(42), -13 + 2 sqrt(42)]
    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-d4a7a90d58b7b7be
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-6 - sqrt(42), 0] to roots in [-13 - 2 sqrt(42), -13 + 2 sqrt(42)]: every PF input with all roots in [-6 - sqrt(42), 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-d4a7a90d58b7b7be
  • 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-5c5351a0cb2ba1df

Proof chains

4 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

No candidate equalities or related sequences.

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