Real-rooted atlas

A144441: Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

A144441 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A144441
\(P_{n}(x) = \left(1 - 3\,x + 4\,n x\right)\,P_{n-1}(x) + \left(4\,x - 4\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
2114116
3183831168
41432155043212416
51218119898198982181144160
6110930217887523548217887109301981184
715467921992371058979510589795219923754679125687424

Show all 15 stored rows

Generated b-files: A144441, A144441 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-723ec2078be12613 · revision 1 · JSON
Definition source
oeis_catalog:A144441
Fingerprint
f6-7729bd9220ade5d3

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,      14,         1;
  3: 1,      83,        83,          1;
  4: 1,     432,      1550,        432,           1;
  5: 1,    2181,     19898,      19898,        2181,           1;
  6: 1,   10930,    217887,     523548,      217887,       10930,          1;
  7: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;
  8: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;
  9: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n-3)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*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 + (4*n - 3)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*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 + (4*n - 3)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*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.

A144441: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,       1;
   3: 1,      14,         1;
   4: 1,      83,        83,          1;
   5: 1,     432,      1550,        432,           1;
   6: 1,    2181,     19898,      19898,        2181,           1;
   7: 1,   10930,    217887,     523548,      217887,       10930,          1;
   8: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;
   9: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;
  10: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*x*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A144441: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,       1;
   3: 1,      14,         1;
   4: 1,      83,        83,          1;
   5: 1,     432,      1550,        432,           1;
   6: 1,    2181,     19898,      19898,        2181,           1;
   7: 1,   10930,    217887,     523548,      217887,       10930,          1;
   8: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;
   9: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;
  10: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*x*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-494af060e65fc342
  • 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-494af060e65fc342
  • 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-494af060e65fc342
  • 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-494af060e65fc342

Proof chains

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

Used in 1 proof step

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