Real-rooted atlas

A038243 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(6 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
23636981
321632416227729
4129625921944648816561
5777619440194409720243024359049
646656139968174960116640437408748729531441
7279936979776146966412247206123601837083061821874782969

Show all 15 stored rows

Generated b-files: A001019 (row sums), A038224 (reversed triangle), A038257, A100062 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-7f05fd9e9c6948d8 · revision 2 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-11aec07ce60b6d61

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:        6,        3;
  2:       36,       36,        9;
  3:      216,      324,      162,        27;
  4:     1296,     2592,     1944,       648,       81;
  5:     7776,    19440,    19440,      9720,     2430,      243;
  6:    46656,   139968,   174960,    116640,    43740,     8748,      729;
  7:   279936,   979776,  1469664,   1224720,   612360,   183708,    30618,    2187;
  8:  1679616,  6718464, 11757312,  11757312,  7348320,  2939328,   734832,  104976,   6561;
  9: 10077696, 45349632, 90699264, 105815808, 79361856, 39680928, 13226976, 2834352, 354294, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (6+3*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[(6 + 3*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] = (6 + 3*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.

A038224: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,      6;
  2:     9,     36,      36;
  3:    27,    162,     324,      216;
  4:    81,    648,    1944,     2592,     1296;
  5:   243,   2430,    9720,    19440,    19440,     7776;
  6:   729,   8748,   43740,   116640,   174960,   139968,     46656;
  7:  2187,  30618,  183708,   612360,  1224720,  1469664,    979776,   279936;
  8:  6561, 104976,  734832,  2939328,  7348320, 11757312,  11757312,  6718464,  1679616;
  9: 19683, 354294, 2834352, 13226976, 39680928, 79361856, 105815808, 90699264, 45349632, 10077696;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+3*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[(6 + 3*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] = (6 + 3*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.

A038257: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,        3;
  2:       36,       36,        9;
  3:      216,      324,      162,        27;
  4:     1296,     2592,     1944,       648,       81;
  5:     7776,    19440,    19440,      9720,     2430,      243;
  6:    46656,   139968,   174960,    116640,    43740,     8748,      729;
  7:   279936,   979776,  1469664,   1224720,   612360,   183708,    30618,    2187;
  8:  1679616,  6718464, 11757312,  11757312,  7348320,  2939328,   734832,  104976,   6561;
  9: 10077696, 45349632, 90699264, 105815808, 79361856, 39680928, 13226976, 2834352, 354294, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (6+3*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[(6 + 3*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] = (6 + 3*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-2d1052dd37eda11b, e-926e24a16f3abf3d
  • 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-2d1052dd37eda11b, e-926e24a16f3abf3d
  • 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-2d1052dd37eda11b, e-926e24a16f3abf3d

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

Same fingerprint (67)

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

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  4. [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