Real-rooted atlas

A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

A356546 palindromic γ-positive nonnegative coefficients only rational roots

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

Recurrence from OEIS A356546
\(\left(n\right)\,P_{n}(x) = \left(-2 - 2\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2612624
320606020160
470280420280701120
525212602520252012602528064
69245544138601848013860554492459136
73432240247207212012012012072072240243432439296

Show all 15 stored rows

Generated b-files: A059304 (row sums), A356546, A356546 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-dd77bf712b1df996 · revision 1 · JSON
Definition source
oeis_catalog:A356546
Fingerprint
f6-77a7ed3b1d5f257e

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): 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:     1;
  1:     2,      2;
  2:     6,     12,       6;
  3:    20,     60,      60,      20;
  4:    70,    280,     420,     280,      70;
  5:   252,   1260,    2520,    2520,    1260,     252;
  6:   924,   5544,   13860,   18480,   13860,    5544,     924;
  7:  3432,  24024,   72072,  120120,  120120,   72072,   24024,    3432;
  8: 12870, 102960,  360360,  720720,  900900,  720720,  360360,  102960,  12870;
  9: 48620, 437580, 1750320, 4084080, 6126120, 6126120, 4084080, 1750320, 437580, 48620;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n-2+(4*n-2)*x)*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[Together[((4*n - 2 + (4*n - 2)*x)*P[n - 1, x])/(n)]];
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] = ((4*n - 2 + (4*n - 2)*x)*v[i - 1])/(n)); 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.

A356546: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     2,      2;
  2:     6,     12,       6;
  3:    20,     60,      60,      20;
  4:    70,    280,     420,     280,      70;
  5:   252,   1260,    2520,    2520,    1260,     252;
  6:   924,   5544,   13860,   18480,   13860,    5544,     924;
  7:  3432,  24024,   72072,  120120,  120120,   72072,   24024,    3432;
  8: 12870, 102960,  360360,  720720,  900900,  720720,  360360,  102960,  12870;
  9: 48620, 437580, 1750320, 4084080, 6126120, 6126120, 4084080, 1750320, 437580, 48620;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n-2+(4*n-2)*x)*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[Together[((4*n - 2 + (4*n - 2)*x)*P[n - 1, x])/(n)]];
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] = ((4*n - 2 + (4*n - 2)*x)*v[i - 1])/(n)); 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.

A356546: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     2,      2;
  2:     6,     12,       6;
  3:    20,     60,      60,      20;
  4:    70,    280,     420,     280,      70;
  5:   252,   1260,    2520,    2520,    1260,     252;
  6:   924,   5544,   13860,   18480,   13860,    5544,     924;
  7:  3432,  24024,   72072,  120120,  120120,   72072,   24024,    3432;
  8: 12870, 102960,  360360,  720720,  900900,  720720,  360360,  102960,  12870;
  9: 48620, 437580, 1750320, 4084080, 6126120, 6126120, 4084080, 1750320, 437580, 48620;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (4*n-2+(4*n-2)*x)*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[Together[((4*n - 2 + (4*n - 2)*x)*P[n - 1, x])/(n)]];
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] = ((4*n - 2 + (4*n - 2)*x)*v[i - 1])/(n)); 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-883c9c594ff729de
  • 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-883c9c594ff729de
  • 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-883c9c594ff729de
  • 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-883c9c594ff729de

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 8 proof steps

Same fingerprint (59)

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