Real-rooted atlas

A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.

A178821 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 A178821
\(\left(n\right)\,P_{n}(x) = \left(4 + 4\,x + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15510
215301560
33510510535280
470280420280701120
5126630126012606301264032
62101260315042003150126021013440
73302310693011550115506930231033042240

Show all 15 stored rows

Generated b-files: A003472 (row sums), A178821, A178821 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-79986ab5023c2764 · revision 2 · JSON
Definition source
oeis_catalog:A178821
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:   5,    5;
  2:  15,   30,    15;
  3:  35,  105,   105,    35;
  4:  70,  280,   420,   280,    70;
  5: 126,  630,  1260,  1260,   630,   126;
  6: 210, 1260,  3150,  4200,  3150,  1260,   210;
  7: 330, 2310,  6930, 11550, 11550,  6930,  2310,   330;
  8: 495, 3960, 13860, 27720, 34650, 27720, 13860,  3960,  495;
  9: 715, 6435, 25740, 60060, 90090, 90090, 60060, 25740, 6435, 715;
%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) = 1.
%t Mathematica
P[0, x_] := 1;
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] = 1; 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.

A178821: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   5,    5;
  2:  15,   30,    15;
  3:  35,  105,   105,    35;
  4:  70,  280,   420,   280,    70;
  5: 126,  630,  1260,  1260,   630,   126;
  6: 210, 1260,  3150,  4200,  3150,  1260,   210;
  7: 330, 2310,  6930, 11550, 11550,  6930,  2310,   330;
  8: 495, 3960, 13860, 27720, 34650, 27720, 13860,  3960,  495;
  9: 715, 6435, 25740, 60060, 90090, 90090, 60060, 25740, 6435, 715;
%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) = 1.
%t Mathematica
P[0, x_] := 1;
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] = 1; 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.

A178821: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   5,    5;
  2:  15,   30,    15;
  3:  35,  105,   105,    35;
  4:  70,  280,   420,   280,    70;
  5: 126,  630,  1260,  1260,   630,   126;
  6: 210, 1260,  3150,  4200,  3150,  1260,   210;
  7: 330, 2310,  6930, 11550, 11550,  6930,  2310,   330;
  8: 495, 3960, 13860, 27720, 34650, 27720, 13860,  3960,  495;
  9: 715, 6435, 25740, 60060, 90090, 90090, 60060, 25740, 6435, 715;
%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) = 1.
%t Mathematica
P[0, x_] := 1;
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] = 1; 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-20c79d8930f25d5b
  • 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-12f9ed6b8198f9ce, e-20c79d8930f25d5b
  • 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-12f9ed6b8198f9ce, e-20c79d8930f25d5b
  • 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-20c79d8930f25d5b
  • 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-12f9ed6b8198f9ce

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