Real-rooted atlas

A027465 → basis A013611

palindromic γ-positive nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
216321664
36419219264512
42561024153610242564096
51024512010240102405120102432768
6409624576614408192061440245764096262144
716384114688344064573440573440344064114688163842097152

Show all 15 stored rows

Generated b-files: A001018 (row sums), A038234, A038234 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-f5249e29b34d4267 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:      4,       4;
  2:     16,      32,      16;
  3:     64,     192,     192,       64;
  4:    256,    1024,    1536,     1024,      256;
  5:   1024,    5120,   10240,    10240,     5120,     1024;
  6:   4096,   24576,   61440,    81920,    61440,    24576,     4096;
  7:  16384,  114688,  344064,   573440,   573440,   344064,   114688,   16384;
  8:  65536,  524288, 1835008,  3670016,  4587520,  3670016,  1835008,  524288,   65536;
  9: 262144, 2359296, 9437184, 22020096, 33030144, 33030144, 22020096, 9437184, 2359296, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (4+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[(4 + 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] = (4 + 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.

A038234: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       4;
  2:     16,      32,      16;
  3:     64,     192,     192,       64;
  4:    256,    1024,    1536,     1024,      256;
  5:   1024,    5120,   10240,    10240,     5120,     1024;
  6:   4096,   24576,   61440,    81920,    61440,    24576,     4096;
  7:  16384,  114688,  344064,   573440,   573440,   344064,   114688,   16384;
  8:  65536,  524288, 1835008,  3670016,  4587520,  3670016,  1835008,  524288,   65536;
  9: 262144, 2359296, 9437184, 22020096, 33030144, 33030144, 22020096, 9437184, 2359296, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (4+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[(4 + 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] = (4 + 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.

A038234: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       4;
  2:     16,      32,      16;
  3:     64,     192,     192,       64;
  4:    256,    1024,    1536,     1024,      256;
  5:   1024,    5120,   10240,    10240,     5120,     1024;
  6:   4096,   24576,   61440,    81920,    61440,    24576,     4096;
  7:  16384,  114688,  344064,   573440,   573440,   344064,   114688,   16384;
  8:  65536,  524288, 1835008,  3670016,  4587520,  3670016,  1835008,  524288,   65536;
  9: 262144, 2359296, 9437184, 22020096, 33030144, 33030144, 22020096, 9437184, 2359296, 262144;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (4+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[(4 + 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] = (4 + 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-b3c62654af8090af, e-e31537aead1e0634
  • 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-b3c62654af8090af, e-e31537aead1e0634
  • 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-b3c62654af8090af, e-e31537aead1e0634

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 (59)

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