Real-rooted atlas

A013609 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(3 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
29241649
32710814464343
4814328647682562401
52431620432057603840102416807
67295832194403456034560184324096117649
7218720412816481814402419201935368601616384823543

Show all 15 stored rows

Generated b-files: A000420 (row sums), A038222, A038233 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-026074a09c74735a · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-c91d5d506c107735

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:     3,      4;
  2:     9,     24,      16;
  3:    27,    108,     144,      64;
  4:    81,    432,     864,     768,     256;
  5:   243,   1620,    4320,    5760,    3840,     1024;
  6:   729,   5832,   19440,   34560,   34560,    18432,    4096;
  7:  2187,  20412,   81648,  181440,  241920,   193536,   86016,   16384;
  8:  6561,  69984,  326592,  870912, 1451520,  1548288, 1032192,  393216,   65536;
  9: 19683, 236196, 1259712, 3919104, 7838208, 10450944, 9289728, 5308416, 1769472, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (3+4*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[(3 + 4*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] = (3 + 4*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.

A038222: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,      4;
  2:     9,     24,      16;
  3:    27,    108,     144,      64;
  4:    81,    432,     864,     768,     256;
  5:   243,   1620,    4320,    5760,    3840,     1024;
  6:   729,   5832,   19440,   34560,   34560,    18432,    4096;
  7:  2187,  20412,   81648,  181440,  241920,   193536,   86016,   16384;
  8:  6561,  69984,  326592,  870912, 1451520,  1548288, 1032192,  393216,   65536;
  9: 19683, 236196, 1259712, 3919104, 7838208, 10450944, 9289728, 5308416, 1769472, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (3+4*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[(3 + 4*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] = (3 + 4*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.

A038233: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       3;
  2:     16,      24,       9;
  3:     64,     144,     108,      27;
  4:    256,     768,     864,     432,       81;
  5:   1024,    3840,    5760,    4320,     1620,     243;
  6:   4096,   18432,   34560,   34560,    19440,    5832,     729;
  7:  16384,   86016,  193536,  241920,   181440,   81648,   20412,    2187;
  8:  65536,  393216, 1032192, 1548288,  1451520,  870912,  326592,   69984,   6561;
  9: 262144, 1769472, 5308416, 9289728, 10450944, 7838208, 3919104, 1259712, 236196, 19683;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+4*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[(3 + 4*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] = (3 + 4*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-95924f9cf72f1e84, e-dd508e760c80720b
  • 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-95924f9cf72f1e84, e-dd508e760c80720b
  • 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-95924f9cf72f1e84, e-dd508e760c80720b

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

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

Same fingerprint (5)

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