Real-rooted atlas

A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).

A085881 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 A085881
\(P_{n}(x) = \left(-1 - x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
236312
315454515120
41054206304201051680
5945472594509450472594530240
610395623701559252079001559256237010395665280
7135135945945283783547297254729725283783594594513513517297280
82027025162162005675670011351340014189175011351340056756700162162002027025518918400
9344594253101348251240539300289459170043418875504341887550289459170012405393003101348253445942517643225600
106547290756547290750294628083757856748900013749310575016499172690013749310575078567489000294628083756547290750654729075670442572800
11137493105751512424163257562120816252268636244875453727248975063521814856506352181485650453727248975022686362448757562120816251512424163251374931057528158588057600
1231623414322537948097187002087145345285069571511509500156535900896375250457441434200292200348339900250457441434200156535900896375695715115095002087145345285037948097187003162341432251295295050649600
137905853580625102776096548125616656579288750226107412405875056526853101468751017483355826437513566444744352500135664447443525001017483355826437556526853101468752261074124058750616656579288750102776096548125790585358062564764752532480000
1421345804667687529884126534762501942468224759562577698728990382500213671504723551875427343009447103750641014514170655625732588016195035000641014514170655625427343009447103750213671504723551875776987289903825001942468224759562529884126534762502134580466768753497296636753920000

Generated b-files: A001813 (row sums), A085881, A085881 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-c7809537965f400d · revision 1 · JSON
Definition source
oeis_catalog:A085881
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:        1,         1;
  2:        3,         6,          3;
  3:       15,        45,         45,         15;
  4:      105,       420,        630,        420,        105;
  5:      945,      4725,       9450,       9450,       4725,        945;
  6:    10395,     62370,     155925,     207900,     155925,      62370,      10395;
  7:   135135,    945945,    2837835,    4729725,    4729725,    2837835,     945945,     135135;
  8:  2027025,  16216200,   56756700,  113513400,  141891750,  113513400,   56756700,   16216200,   2027025;
  9: 34459425, 310134825, 1240539300, 2894591700, 4341887550, 4341887550, 2894591700, 1240539300, 310134825, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(2*n-1)*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[(2*n - 1 + (2*n - 1)*x)*P[n - 1, 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] = (2*n - 1 + (2*n - 1)*x)*v[i - 1]); 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.

A085881: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        1,         1;
  2:        3,         6,          3;
  3:       15,        45,         45,         15;
  4:      105,       420,        630,        420,        105;
  5:      945,      4725,       9450,       9450,       4725,        945;
  6:    10395,     62370,     155925,     207900,     155925,      62370,      10395;
  7:   135135,    945945,    2837835,    4729725,    4729725,    2837835,     945945,     135135;
  8:  2027025,  16216200,   56756700,  113513400,  141891750,  113513400,   56756700,   16216200,   2027025;
  9: 34459425, 310134825, 1240539300, 2894591700, 4341887550, 4341887550, 2894591700, 1240539300, 310134825, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(2*n-1)*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[(2*n - 1 + (2*n - 1)*x)*P[n - 1, 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] = (2*n - 1 + (2*n - 1)*x)*v[i - 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.

A085881: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        1,         1;
  2:        3,         6,          3;
  3:       15,        45,         45,         15;
  4:      105,       420,        630,        420,        105;
  5:      945,      4725,       9450,       9450,       4725,        945;
  6:    10395,     62370,     155925,     207900,     155925,      62370,      10395;
  7:   135135,    945945,    2837835,    4729725,    4729725,    2837835,     945945,     135135;
  8:  2027025,  16216200,   56756700,  113513400,  141891750,  113513400,   56756700,   16216200,   2027025;
  9: 34459425, 310134825, 1240539300, 2894591700, 4341887550, 4341887550, 2894591700, 1240539300, 310134825, 34459425;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2*n-1+(2*n-1)*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[(2*n - 1 + (2*n - 1)*x)*P[n - 1, 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] = (2*n - 1 + (2*n - 1)*x)*v[i - 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 >= 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-1d211cea19207c02
  • 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-088407700c450d53, e-1d211cea19207c02
  • 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-088407700c450d53, e-1d211cea19207c02
  • 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-1d211cea19207c02
  • 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-088407700c450d53

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 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