Real-rooted atlas

A030528: Triangle read by rows: T(n,k) = binomial(k,n-k).

A030528 nonnegative coefficients

Proved for all n: all roots real, in (−4, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A030528
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
20213
301315
4003418
500165113
60004106121
7000110157134

Show all 15 stored rows

Generated b-files: A030528, A169803 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Basis-map coefficients of A030528, shifted · proved equal for all n (proof step)

\(\left(1 + n\right)\,P_{n}(x) = \left(4 + 2\,x - 2\,n\right)\,P_{n-1}(x) + \left(4\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-21854f4257effb01 · revision 1 · JSON
Definition source
oeis_catalog:A030528
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A030528 (equal by e-33efc42fb8feded5)
Fingerprint
f6-8de4b75c4559d58b

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: no
  • γ-positive (gamma_positive): no
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1, 1;
  2: 0, 2, 1;
  3: 0, 1, 3, 1;
  4: 0, 0, 3, 4,  1;
  5: 0, 0, 1, 6,  5,  1;
  6: 0, 0, 0, 4, 10,  6,  1;
  7: 0, 0, 0, 1, 10, 15,  7,  1;
  8: 0, 0, 0, 0,  5, 20, 21,  8, 1;
  9: 0, 0, 0, 0,  1, 15, 35, 28, 9, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + x*v[i - 2]); 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.

A030528: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1, 1;
   3: 0, 2, 1;
   4: 0, 1, 3, 1;
   5: 0, 0, 3, 4,  1;
   6: 0, 0, 1, 6,  5,  1;
   7: 0, 0, 0, 4, 10,  6,  1;
   8: 0, 0, 0, 1, 10, 15,  7,  1;
   9: 0, 0, 0, 0,  5, 20, 21,  8, 1;
  10: 0, 0, 0, 0,  1, 15, 35, 28, 9, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = x*v[i - 1] + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A169803: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1, 1;
  2: 1, 2,  0;
  3: 1, 3,  1,  0;
  4: 1, 4,  3,  0,  0;
  5: 1, 5,  6,  1,  0, 0;
  6: 1, 6, 10,  4,  0, 0, 0;
  7: 1, 7, 15, 10,  1, 0, 0, 0;
  8: 1, 8, 21, 20,  5, 0, 0, 0, 0;
  9: 1, 9, 28, 35, 15, 1, 0, 0, 0, 0;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + x*v[i - 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(9)

Generated with polytool c48629374597.

Proved properties

Proof chains

8 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 18 proof steps

Same fingerprint (5)

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741