Real-rooted atlas

A142458: Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).

A142458 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A142458
\(P_{n}(x) = \left(1 - 2\,x + 3\,n x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
218110
313939180
411665461661880
5167754825482677112320
612724471751096404717527241209440
7110915373809170967517096753738091091514188800

Show all 15 stored rows

Generated b-files: A008544 (row sums), A142458, A142458 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-ddb0cd3a4d6edf09 · revision 1 · JSON
Definition source
oeis_catalog:A142458
Fingerprint
f6-257640823bee26bf

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,     39,       39,         1;
  4: 1,    166,      546,       166,         1;
  5: 1,    677,     5482,      5482,       677,         1;
  6: 1,   2724,    47175,    109640,     47175,      2724,         1;
  7: 1,  10915,   373809,   1709675,   1709675,    373809,     10915,        1;
  8: 1,  43682,  2824048,  23077694,  44451550,  23077694,   2824048,    43682,      1;
  9: 1, 174753, 20729340, 284433852, 947113254, 947113254, 284433852, 20729340, 174753, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n-2)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n - 2)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (3*n - 2)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); 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.

A142458: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,      8,        1;
   4: 1,     39,       39,         1;
   5: 1,    166,      546,       166,         1;
   6: 1,    677,     5482,      5482,       677,         1;
   7: 1,   2724,    47175,    109640,     47175,      2724,         1;
   8: 1,  10915,   373809,   1709675,   1709675,    373809,     10915,        1;
   9: 1,  43682,  2824048,  23077694,  44451550,  23077694,   2824048,    43682,      1;
  10: 1, 174753, 20729340, 284433852, 947113254, 947113254, 284433852, 20729340, 174753, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n-5)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n - 5)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], 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; for(n = 2, N, my(i = n); v[i] = (1 + (3*n - 5)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); 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.

A142458: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,      8,        1;
   4: 1,     39,       39,         1;
   5: 1,    166,      546,       166,         1;
   6: 1,    677,     5482,      5482,       677,         1;
   7: 1,   2724,    47175,    109640,     47175,      2724,         1;
   8: 1,  10915,   373809,   1709675,   1709675,    373809,     10915,        1;
   9: 1,  43682,  2824048,  23077694,  44451550,  23077694,   2824048,    43682,      1;
  10: 1, 174753, 20729340, 284433852, 947113254, 947113254, 284433852, 20729340, 174753, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(3*n-5)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n - 5)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], 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; for(n = 2, N, my(i = n); v[i] = (1 + (3*n - 5)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); 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-98e347ce1f066bf3
  • 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-98e347ce1f066bf3
  • 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-98e347ce1f066bf3
  • 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-98e347ce1f066bf3

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 3 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