Real-rooted atlas

A028338: Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.

A028338 nonnegative coefficients only rational roots

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

Recurrence from OEIS A028338
\(P_{n}(x) = \left(-1 + x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
23418
315239148
410517686161384
594516899502302513840
6103951952412139348050536146080
71351352642071773315737910045973491645120

Show all 15 stored rows

Generated b-files: A000165 (row sums), A028338, A109692 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-4f6e747b794bcc2e · revision 1 · JSON
Definition source
oeis_catalog:A028338
Fingerprint
f6-f8baf88ce86e36d7

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): 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:        1,        1;
  2:        3,        4,        1;
  3:       15,       23,        9,        1;
  4:      105,      176,       86,       16,       1;
  5:      945,     1689,      950,      230,      25,      1;
  6:    10395,    19524,    12139,     3480,     505,     36,     1;
  7:   135135,   264207,   177331,    57379,   10045,    973,    49,    1;
  8:  2027025,  4098240,  2924172,  1038016,  208054,  24640,  1708,   64,  1;
  9: 34459425, 71697105, 53809164, 20570444, 4574934, 626934, 53676, 2796, 81, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+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[(2*n - 1 + x)*P[n - 1, x]];
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] = (2*n - 1 + x)*v[i - 1]); 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.

A028338: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        1,        1;
  2:        3,        4,        1;
  3:       15,       23,        9,        1;
  4:      105,      176,       86,       16,       1;
  5:      945,     1689,      950,      230,      25,      1;
  6:    10395,    19524,    12139,     3480,     505,     36,     1;
  7:   135135,   264207,   177331,    57379,   10045,    973,    49,    1;
  8:  2027025,  4098240,  2924172,  1038016,  208054,  24640,  1708,   64,  1;
  9: 34459425, 71697105, 53809164, 20570444, 4574934, 626934, 53676, 2796, 81, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+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[(2*n - 1 + x)*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + x)*v[i - 1]); 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.

A109692: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,    3;
  3: 1,  9,   23,    15;
  4: 1, 16,   86,   176,    105;
  5: 1, 25,  230,   950,   1689,     945;
  6: 1, 36,  505,  3480,  12139,   19524,    10395;
  7: 1, 49,  973, 10045,  57379,  177331,   264207,   135135;
  8: 1, 64, 1708, 24640, 208054, 1038016,  2924172,  4098240,  2027025;
  9: 1, 81, 2796, 53676, 626934, 4574934, 20570444, 53809164, 71697105, 34459425;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2*n-1+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[(2*n - 1 + x)*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + x)*v[i - 1]); 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-59b849691f316c6f
  • 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-59b849691f316c6f, e-ef36674c01230a86
  • 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-59b849691f316c6f, e-ef36674c01230a86
  • 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-59b849691f316c6f
  • 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-ef36674c01230a86
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: (-inf, 0] -> (-inf, 0] through R_s, s = 2, and U_g
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-7739b02859f35716

Proof chains

3 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 25 proof steps

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

References

  1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
  2. [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
  3. [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