Real-rooted atlas

A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).

A344565 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 A344565
\(\left(n\right)\,P_{n}(x) = \left(4 + 4\,x + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
033
1151530
2459045180
3105315315105840
421084012608402103360
5378189037803780189037812096
663037809450126009450378063040320
79906930207903465034650207906930990126720
814851188041580831601039508316041580118801485380160
921451930577220180180270270270270180180772201930521451098240
103003300301351353603606306307567566306303603601351353003030033075072
1140954504522522567567513513501891890189189013513506756752252254504540958386560
12546065520360360120120027027004324320504504043243202702700120120036036065520546022364160
13714092820556920204204051051009189180122522401225224091891805105100204204055692092820714058490880
149180128520835380334152091891801837836027567540315057602756754018378360918918033415208353801285209180150405120

Generated b-files: A344564 (row sums), A344565, A344565 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-dd4aa31f3d11d152 · revision 2 · JSON
Definition source
oeis_catalog:A344565
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:    3;
  1:   15,    15;
  2:   45,    90,    45;
  3:  105,   315,   315,    105;
  4:  210,   840,  1260,    840,    210;
  5:  378,  1890,  3780,   3780,   1890,    378;
  6:  630,  3780,  9450,  12600,   9450,   3780,    630;
  7:  990,  6930, 20790,  34650,  34650,  20790,   6930,   990;
  8: 1485, 11880, 41580,  83160, 103950,  83160,  41580, 11880,  1485;
  9: 2145, 19305, 77220, 180180, 270270, 270270, 180180, 77220, 19305, 2145;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+4+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n + 4)*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] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n + 4)*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.

A344565: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    3;
  1:   15,    15;
  2:   45,    90,    45;
  3:  105,   315,   315,    105;
  4:  210,   840,  1260,    840,    210;
  5:  378,  1890,  3780,   3780,   1890,    378;
  6:  630,  3780,  9450,  12600,   9450,   3780,    630;
  7:  990,  6930, 20790,  34650,  34650,  20790,   6930,   990;
  8: 1485, 11880, 41580,  83160, 103950,  83160,  41580, 11880,  1485;
  9: 2145, 19305, 77220, 180180, 270270, 270270, 180180, 77220, 19305, 2145;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+4+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n + 4)*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] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n + 4)*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.

A344565: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    3;
  1:   15,    15;
  2:   45,    90,    45;
  3:  105,   315,   315,    105;
  4:  210,   840,  1260,    840,    210;
  5:  378,  1890,  3780,   3780,   1890,    378;
  6:  630,  3780,  9450,  12600,   9450,   3780,    630;
  7:  990,  6930, 20790,  34650,  34650,  20790,   6930,   990;
  8: 1485, 11880, 41580,  83160, 103950,  83160,  41580, 11880,  1485;
  9: 2145, 19305, 77220, 180180, 270270, 270270, 180180, 77220, 19305, 2145;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (n+4+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n + 4)*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] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n + 4)*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-e2cde2e22920ce64
  • 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-4dae17a76888d380, e-e2cde2e22920ce64
  • 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-4dae17a76888d380, e-e2cde2e22920ce64
  • 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-e2cde2e22920ce64
  • 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-4dae17a76888d380

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 17 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