Real-rooted atlas

A013611 → basis A007318

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
225401681
312530024064729
46252000240012802566561
531251250020000160006400102459049
6156257500015000016000096000307204096531441
778125437500105000014000001120000537600143360163844782969

Show all 15 stored rows

Generated b-files: A001019 (row sums), A013628 (reversed triangle), A038246, A100062 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-d04994959b21388c · revision 2 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-6b726760bd05165a

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:       5,        4;
  2:      25,       40,       16;
  3:     125,      300,      240,       64;
  4:     625,     2000,     2400,     1280,       256;
  5:    3125,    12500,    20000,    16000,      6400,     1024;
  6:   15625,    75000,   150000,   160000,     96000,    30720,     4096;
  7:   78125,   437500,  1050000,  1400000,   1120000,   537600,   143360,    16384;
  8:  390625,  2500000,  7000000, 11200000,  11200000,  7168000,  2867200,   655360,   65536;
  9: 1953125, 14062500, 45000000, 84000000, 100800000, 80640000, 43008000, 14745600, 2949120, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (5+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[(5 + 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] = (5 + 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.

A013628: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       5;
  2:     16,      40,       25;
  3:     64,     240,      300,      125;
  4:    256,    1280,     2400,     2000,      625;
  5:   1024,    6400,    16000,    20000,    12500,      3125;
  6:   4096,   30720,    96000,   160000,   150000,     75000,    15625;
  7:  16384,  143360,   537600,  1120000,  1400000,   1050000,   437500,    78125;
  8:  65536,  655360,  2867200,  7168000, 11200000,  11200000,  7000000,  2500000,   390625;
  9: 262144, 2949120, 14745600, 43008000, 80640000, 100800000, 84000000, 45000000, 14062500, 1953125;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+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[(5 + 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] = (5 + 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.

A038246: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,        4;
  2:      25,       40,       16;
  3:     125,      300,      240,       64;
  4:     625,     2000,     2400,     1280,       256;
  5:    3125,    12500,    20000,    16000,      6400,     1024;
  6:   15625,    75000,   150000,   160000,     96000,    30720,     4096;
  7:   78125,   437500,  1050000,  1400000,   1120000,   537600,   143360,    16384;
  8:  390625,  2500000,  7000000, 11200000,  11200000,  7168000,  2867200,   655360,   65536;
  9: 1953125, 14062500, 45000000, 84000000, 100800000, 80640000, 43008000, 14745600, 2949120, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (5+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[(5 + 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] = (5 + 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.

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-13c78408b476aecd, e-a0d290e16ed9f688, e-ac61fd7aa24ce6bb
  • 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-13c78408b476aecd, e-a0d290e16ed9f688, e-ac61fd7aa24ce6bb, e-f1a8fc67b53c2d66
  • 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-13c78408b476aecd, e-a0d290e16ed9f688, e-ac61fd7aa24ce6bb, e-f1a8fc67b53c2d66
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-f1a8fc67b53c2d66
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-f1a8fc67b53c2d66

Proof chains

4 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 (6)

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