Real-rooted atlas

A375853: Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.

A375853 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 A375853
\(\left(1 - n\right)\,P_{n}(x) = \left(-2 + 2\,x - n - 3\,n x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
122
28816
320562096
44021621640512
5706161188616702560
6112145645764576145611212288
71683024140402288014040302416857344
82405712367208840088400367205712240262144
9330100328527228424041990028424085272100323301179648
104401663218057679587216279201627920795872180576166324405242880
1157226312355212199971254083127488432540831219997123552122631257223068672
1272840040657800460460015906800291216802912168015906800460460065780040040728100663296
139105896811583009867000423832509911160013104756099111600423832509867000115830058968910436207616
141120844481954368199056001040520003025512005109753605109753603025512001040520001990560019543688444811201879048192
15136011804831788643814636823841480084362160017716053602262890880177160536084362160023841480038146368317886411804813608053063680

Generated b-files: A375853, A375853 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-92be85f47bd42a26 · revision 1 · JSON
Definition source
oeis_catalog:A375853
Fingerprint
f6-f0bc0625362e6d07

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): 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   2;
   2:   8,     8;
   3:  20,    56,     20;
   4:  40,   216,    216,     40;
   5:  70,   616,   1188,    616,      70;
   6: 112,  1456,   4576,   4576,    1456,     112;
   7: 168,  3024,  14040,  22880,   14040,    3024,    168;
   8: 240,  5712,  36720,  88400,   88400,   36720,   5712,    240;
   9: 330, 10032,  85272, 284240,  419900,  284240,  85272,  10032,   330;
  10: 440, 16632, 180576, 795872, 1627920, 1627920, 795872, 180576, 16632, 440;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = -(n+2+(3*n-2)*x)*P(n-1,x) - (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 2 + (3*n - 2)*x)*P[n - 1, x] - (2*x - 2*x^2)*D[P[n - 1, x], x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = (-(n + 2 + (3*n - 2)*x)*v[i - 1] - (2*x - 2*x^2)*deriv(v[i - 1], x))/(-n + 1)); 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.

A375853: triangle, offset 2, from the row P1
%e Example
Triangle begins:
   2:   2;
   3:   8,     8;
   4:  20,    56,     20;
   5:  40,   216,    216,     40;
   6:  70,   616,   1188,    616,      70;
   7: 112,  1456,   4576,   4576,    1456,     112;
   8: 168,  3024,  14040,  22880,   14040,    3024,    168;
   9: 240,  5712,  36720,  88400,   88400,   36720,   5712,    240;
  10: 330, 10032,  85272, 284240,  419900,  284240,  85272,  10032,   330;
  11: 440, 16632, 180576, 795872, 1627920, 1627920, 795872, 180576, 16632, 440;
%F Formula
The row polynomials satisfy -(n-2)*P(n,x) = -(n+1+(3*n-5)*x)*P(n-1,x) - (2*x-2*x^2)*d/dx P(n-1,x) for n >= 3, with P(2,x) = 2.
%t Mathematica
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (3*n - 5)*x)*P[n - 1, x] - (2*x - 2*x^2)*D[P[n - 1, x], x])/(-n + 2)]];
Table[PadRight[CoefficientList[P[n, x], x], n - 1], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 2; for(n = 3, N, my(i = n - 1); v[i] = (-(n + 1 + (3*n - 5)*x)*v[i - 1] - (2*x - 2*x^2)*deriv(v[i - 1], x))/(-n + 2)); 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(11)

Generated with polytool c48629374597.

A375853: reversed triangle, offset 2, from the row P1
%e Example
Triangle begins:
   2:   2;
   3:   8,     8;
   4:  20,    56,     20;
   5:  40,   216,    216,     40;
   6:  70,   616,   1188,    616,      70;
   7: 112,  1456,   4576,   4576,    1456,     112;
   8: 168,  3024,  14040,  22880,   14040,    3024,    168;
   9: 240,  5712,  36720,  88400,   88400,   36720,   5712,    240;
  10: 330, 10032,  85272, 284240,  419900,  284240,  85272,  10032,   330;
  11: 440, 16632, 180576, 795872, 1627920, 1627920, 795872, 180576, 16632, 440;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where -(n-2)*P(n,x) = -(n+1+(3*n-5)*x)*P(n-1,x) - (2*x-2*x^2)*d/dx P(n-1,x) for n >= 3, with P(2,x) = 2.
%t Mathematica
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (3*n - 5)*x)*P[n - 1, x] - (2*x - 2*x^2)*D[P[n - 1, x], x])/(-n + 2)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 1]], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 2; for(n = 3, N, my(i = n - 1); v[i] = (-(n + 1 + (3*n - 5)*x)*v[i - 1] - (2*x - 2*x^2)*deriv(v[i - 1], x))/(-n + 2)); 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(11)

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-b3e0ac60e2e16f0a
  • 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-b3e0ac60e2e16f0a
  • 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-b3e0ac60e2e16f0a
  • 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-b3e0ac60e2e16f0a

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 2 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