Real-rooted atlas

A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.

A026729 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 A026729
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30112
400213
5001315
60003418
7000165113
800004106121

Show all 15 stored rows

Generated b-files: A011973 (reversed triangle), A026729, A052553 (reversed triangle), A098925 ?

Computed in your browser ?

Other recurrences

Basis-map coefficients of A026729, shifted · proved equal for all n (proof step)

\(\left(-1 + n\right)\,P_{n}(x) = \left(4 + x - 2\,n\right)\,P_{n-1}(x) + \left(4\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-392f49b496f79fbf · revision 1 · JSON
Definition source
oeis_catalog:A026729
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A026729 (equal by e-51b17c42943e3651)
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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 1, 1;
   4: 0, 0, 2, 1;
   5: 0, 0, 1, 3, 1;
   6: 0, 0, 0, 3, 4,  1;
   7: 0, 0, 0, 1, 6,  5,  1;
   8: 0, 0, 0, 0, 4, 10,  6,  1;
   9: 0, 0, 0, 0, 1, 10, 15,  7, 1;
  10: 0, 0, 0, 0, 0,  5, 20, 21, 8, 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) = x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 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};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A011973: reversed triangle, offset 0, from the row P1, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1, 1;
  3: 1, 2;
  4: 1, 3,  1;
  5: 1, 4,  3;
  6: 1, 5,  6,  1;
  7: 1, 6, 10,  4;
  8: 1, 7, 15, 10, 1;
  9: 1, 8, 21, 20, 5;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, 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) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 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/x^valuation(p, x))); 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.

A026729: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 1;
  3: 0, 0, 2, 1;
  4: 0, 0, 1, 3, 1;
  5: 0, 0, 0, 3, 4,  1;
  6: 0, 0, 0, 1, 6,  5,  1;
  7: 0, 0, 0, 0, 4, 10,  6,  1;
  8: 0, 0, 0, 0, 1, 10, 15,  7, 1;
  9: 0, 0, 0, 0, 0,  5, 20, 21, 8, 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) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 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 + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 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};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A052553: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1, 0;
  2: 1, 1,  0;
  3: 1, 2,  0,  0;
  4: 1, 3,  1,  0, 0;
  5: 1, 4,  3,  0, 0, 0;
  6: 1, 5,  6,  1, 0, 0, 0;
  7: 1, 6, 10,  4, 0, 0, 0, 0;
  8: 1, 7, 15, 10, 1, 0, 0, 0, 0;
  9: 1, 8, 21, 20, 5, 0, 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) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 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] = 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.

A098925: triangle, offset 0, from the row P1, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1,  1;
  3: 2,  1;
  4: 1,  3,  1;
  5: 3,  4,  1;
  6: 1,  6,  5, 1;
  7: 4, 10,  6, 1;
  8: 1, 10, 15, 7, 1;
  9: 5, 20, 21, 8, 1;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, 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) = x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, x]];
Table[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 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/x^valuation(p, x))); 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 >= 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-24564bef34b2c2e5, e-fcd19baf8f835cf0
  • 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-0178a3d1cd265fb6, e-24564bef34b2c2e5, e-fa68694d112454ed, e-fcd19baf8f835cf0
  • 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-0178a3d1cd265fb6, e-24564bef34b2c2e5, e-fa68694d112454ed, e-fcd19baf8f835cf0
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [-4, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-fa68694d112454ed
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0178a3d1cd265fb6
  • 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-0178a3d1cd265fb6

Proof chains

5 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 16 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