Real-rooted atlas

A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

A085880 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 A085880
\(\left(n\right)\,P_{n}(x) = \left(-6 - 6\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
32428
451515540
51456845614224
642210420420210421344
71327921980264019807921328448
84293003900915015150159009300342954912

Show all 15 stored rows

Generated b-files: A085880, A085880 (reversed triangle), A151374 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-a98fae1bf9eeb913 · revision 2 · JSON
Definition source
oeis_catalog:A085880
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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    1,     1;
   3:    2,     4,      2;
   4:    5,    15,     15,      5;
   5:   14,    56,     84,     56,     14;
   6:   42,   210,    420,    420,    210,     42;
   7:  132,   792,   1980,   2640,   1980,    792,    132;
   8:  429,  3003,   9009,  15015,  15015,   9009,   3003,    429;
   9: 1430, 11440,  40040,  80080, 100100,  80080,  40040,  11440,  1430;
  10: 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n-6+(4*n-6)*x)*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[Together[((4*n - 6 + (4*n - 6)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {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] = ((4*n - 6 + (4*n - 6)*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A085880: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:    1;
  1:    1,     1;
  2:    2,     4,      2;
  3:    5,    15,     15,      5;
  4:   14,    56,     84,     56,     14;
  5:   42,   210,    420,    420,    210,     42;
  6:  132,   792,   1980,   2640,   1980,    792,    132;
  7:  429,  3003,   9009,  15015,  15015,   9009,   3003,    429;
  8: 1430, 11440,  40040,  80080, 100100,  80080,  40040,  11440,  1430;
  9: 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;
%F Formula
The row polynomials satisfy (n+1)*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 + 1)]];
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 + 1)); 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.

A085880: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:    1;
  1:    1,     1;
  2:    2,     4,      2;
  3:    5,    15,     15,      5;
  4:   14,    56,     84,     56,     14;
  5:   42,   210,    420,    420,    210,     42;
  6:  132,   792,   1980,   2640,   1980,    792,    132;
  7:  429,  3003,   9009,  15015,  15015,   9009,   3003,    429;
  8: 1430, 11440,  40040,  80080, 100100,  80080,  40040,  11440,  1430;
  9: 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*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 + 1)]];
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 + 1)); 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 >= 1, 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-68a5cf53bdad0f8f
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-68a5cf53bdad0f8f, e-93440eb5a87a1e1b
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-68a5cf53bdad0f8f, e-93440eb5a87a1e1b
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-68a5cf53bdad0f8f
  • Real-rooted. Every row P_n, n >= 1, 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-93440eb5a87a1e1b

Proof chains

2 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

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