Real-rooted atlas

A038208 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(4 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2166464144
3643847685121728
4256204861448192409620736
510241024040960819208192032768248832
64096491522457606553609830407864322621442985984
716384229376137625645875209175040110100487340032209715235831808

Show all 15 stored rows

Generated b-files: A001021 (row sums), A038238, A038282 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-0be19de8f4a58a03 · revision 1 · JSON
Definition source
derived:basis_map_A013611
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:      4,       8;
  2:     16,      64,       64;
  3:     64,     384,      768,       512;
  4:    256,    2048,     6144,      8192,      4096;
  5:   1024,   10240,    40960,     81920,     81920,      32768;
  6:   4096,   49152,   245760,    655360,    983040,     786432,     262144;
  7:  16384,  229376,  1376256,   4587520,   9175040,   11010048,    7340032,    2097152;
  8:  65536, 1048576,  7340032,  29360128,  73400320,  117440512,  117440512,   67108864,  16777216;
  9: 262144, 4718592, 37748736, 176160768, 528482304, 1056964608, 1409286144, 1207959552, 603979776, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (4+8*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 + 8*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 + 8*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.

A038238: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       8;
  2:     16,      64,       64;
  3:     64,     384,      768,       512;
  4:    256,    2048,     6144,      8192,      4096;
  5:   1024,   10240,    40960,     81920,     81920,      32768;
  6:   4096,   49152,   245760,    655360,    983040,     786432,     262144;
  7:  16384,  229376,  1376256,   4587520,   9175040,   11010048,    7340032,    2097152;
  8:  65536, 1048576,  7340032,  29360128,  73400320,  117440512,  117440512,   67108864,  16777216;
  9: 262144, 4718592, 37748736, 176160768, 528482304, 1056964608, 1409286144, 1207959552, 603979776, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (4+8*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 + 8*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 + 8*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.

A038282: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         8,         4;
  2:        64,        64,         16;
  3:       512,       768,        384,         64;
  4:      4096,      8192,       6144,       2048,        256;
  5:     32768,     81920,      81920,      40960,      10240,      1024;
  6:    262144,    786432,     983040,     655360,     245760,     49152,      4096;
  7:   2097152,   7340032,   11010048,    9175040,    4587520,   1376256,    229376,    16384;
  8:  16777216,  67108864,  117440512,  117440512,   73400320,  29360128,   7340032,  1048576,   65536;
  9: 134217728, 603979776, 1207959552, 1409286144, 1056964608, 528482304, 176160768, 37748736, 4718592, 262144;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (4+8*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 + 8*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 + 8*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-e6fcabb7b5383fc4
  • 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-e6fcabb7b5383fc4
  • 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-e6fcabb7b5383fc4

Proof chains

1 incoming proof step. 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