Real-rooted atlas

A038323: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.

A038323 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 A038323
\(P_{n}(x) = \left(11 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111920
212119881400
31331326726737298000
4146414791658806320766561160000
51610516588451078110882090360855590493200000
617715618696754177888151940598011908215389723453144164000000
7194871711116083432739477513735651153056441851500435094092095747829691280000000

Show all 15 stored rows

Generated b-files: A009964 (row sums), A038301 (reversed triangle), A038323 ?

Computed in your browser ?

Technical details
Node id
n-4e096b5929fd3f2e · revision 1 · JSON
Definition source
oeis_catalog:A038323
Fingerprint
f6-6bae7ffb6b8d3725

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:         11,           9;
  2:        121,         198,          81;
  3:       1331,        3267,        2673,          729;
  4:      14641,       47916,       58806,        32076,         6561;
  5:     161051,      658845,     1078110,       882090,       360855,        59049;
  6:    1771561,     8696754,    17788815,     19405980,     11908215,      3897234,      531441;
  7:   19487171,   111608343,   273947751,    373565115,    305644185,    150043509,    40920957,     4782969;
  8:  214358881,  1403076312,  4017900348,   6574746024,   6724172070,   4401276264,  1800522108,   420901272,   43046721;
  9: 2357947691, 17363069361, 56824590636, 108483309396, 133138606986, 108931587534, 59417229564, 20834612964, 4261625379, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (11+9*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[(11 + 9*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] = (11 + 9*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.

A038301: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         9,         11;
  2:        81,        198,         121;
  3:       729,       2673,        3267,        1331;
  4:      6561,      32076,       58806,       47916,        14641;
  5:     59049,     360855,      882090,     1078110,       658845,       161051;
  6:    531441,    3897234,    11908215,    19405980,     17788815,      8696754,      1771561;
  7:   4782969,   40920957,   150043509,   305644185,    373565115,    273947751,    111608343,    19487171;
  8:  43046721,  420901272,  1800522108,  4401276264,   6724172070,   6574746024,   4017900348,  1403076312,   214358881;
  9: 387420489, 4261625379, 20834612964, 59417229564, 108931587534, 133138606986, 108483309396, 56824590636, 17363069361, 2357947691;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+9*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[(11 + 9*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] = (11 + 9*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.

A038323: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,           9;
  2:        121,         198,          81;
  3:       1331,        3267,        2673,          729;
  4:      14641,       47916,       58806,        32076,         6561;
  5:     161051,      658845,     1078110,       882090,       360855,        59049;
  6:    1771561,     8696754,    17788815,     19405980,     11908215,      3897234,      531441;
  7:   19487171,   111608343,   273947751,    373565115,    305644185,    150043509,    40920957,     4782969;
  8:  214358881,  1403076312,  4017900348,   6574746024,   6724172070,   4401276264,  1800522108,   420901272,   43046721;
  9: 2357947691, 17363069361, 56824590636, 108483309396, 133138606986, 108931587534, 59417229564, 20834612964, 4261625379, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (11+9*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[(11 + 9*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] = (11 + 9*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.

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-3b00b07e8aa23801
  • 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-3b00b07e8aa23801, e-896c34b831f8e9e4
  • 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-3b00b07e8aa23801, e-896c34b831f8e9e4
  • 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-3b00b07e8aa23801
  • 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-896c34b831f8e9e4
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times 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-de1df13ef3432c36
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 11] to roots in (-inf, 0]: an input root r <= 11 gives the root (r - a)/d <= 0; a larger root gives a positive root
    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-de1df13ef3432c36

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 7 proof steps

Same fingerprint (3)

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