Real-rooted atlas

A257621: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.

A257621 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 A257621
\(P_{n}(x) = \left(3 - x + 4\,n x\right)\,P_{n-1}(x) + \left(4\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2942960
32739339327840
4813156864631568115120
52432363114244614244623631243332640
67291710062015895427338020158951710067298648640
721871216725261079831024027051024027052610798312167252187259459200

Show all 15 stored rows

Generated b-files: A000407 (row sums), A257621, A257621 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-930be05ae3b321d9 · revision 1 · JSON
Definition source
oeis_catalog:A257621
Fingerprint
f6-536e72d45fb98f2a

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:     3,        3;
  2:     9,       42,          9;
  3:    27,      393,        393,          27;
  4:    81,     3156,       8646,        3156,           81;
  5:   243,    23631,     142446,      142446,        23631,          243;
  6:   729,   171006,    2015895,     4273380,      2015895,       171006,         729;
  7:  2187,  1216725,   26107983,   102402705,    102402705,     26107983,     1216725,       2187;
  8:  6561,  8584872,  320039388,  2136524184,   3891302790,   2136524184,   320039388,    8584872,     6561;
  9: 19683, 60323739, 3786564300, 40688926236, 123074809242, 123074809242, 40688926236, 3786564300, 60323739, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (3+(4*n-1)*x)*P(n-1,x) + (4*x-4*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[(3 + (4*n - 1)*x)*P[n - 1, x] + (4*x - 4*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] = (3 + (4*n - 1)*x)*v[i - 1] + (4*x - 4*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.

A257621: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,        3;
  2:     9,       42,          9;
  3:    27,      393,        393,          27;
  4:    81,     3156,       8646,        3156,           81;
  5:   243,    23631,     142446,      142446,        23631,          243;
  6:   729,   171006,    2015895,     4273380,      2015895,       171006,         729;
  7:  2187,  1216725,   26107983,   102402705,    102402705,     26107983,     1216725,       2187;
  8:  6561,  8584872,  320039388,  2136524184,   3891302790,   2136524184,   320039388,    8584872,     6561;
  9: 19683, 60323739, 3786564300, 40688926236, 123074809242, 123074809242, 40688926236, 3786564300, 60323739, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (3+(4*n-1)*x)*P(n-1,x) + (4*x-4*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[(3 + (4*n - 1)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (3 + (4*n - 1)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x)); 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.

A257621: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,        3;
  2:     9,       42,          9;
  3:    27,      393,        393,          27;
  4:    81,     3156,       8646,        3156,           81;
  5:   243,    23631,     142446,      142446,        23631,          243;
  6:   729,   171006,    2015895,     4273380,      2015895,       171006,         729;
  7:  2187,  1216725,   26107983,   102402705,    102402705,     26107983,     1216725,       2187;
  8:  6561,  8584872,  320039388,  2136524184,   3891302790,   2136524184,   320039388,    8584872,     6561;
  9: 19683, 60323739, 3786564300, 40688926236, 123074809242, 123074809242, 40688926236, 3786564300, 60323739, 19683;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+(4*n-1)*x)*P(n-1,x) + (4*x-4*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[(3 + (4*n - 1)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (3 + (4*n - 1)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x)); 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-53870c9c687f3bf2
  • 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-53870c9c687f3bf2
  • 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-53870c9c687f3bf2
  • 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-53870c9c687f3bf2

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

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