Real-rooted atlas

A013611 → 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
\(P_{n}(x) = \left(5 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151621
225160256441
31251200384040969261
46258000384008192065536194481
53125500003200001024000163840010485764084101
61562530000024000001024000024576000314572801677721685766121
778125175000016800000896000002867200005505024005872025602684354561801088541
839062510000000112000000716800000286720000073400320001174405120010737418240429496729637822859361
91953125562500007200000005376000000258048000008257536000017616076800024159191040019327352832068719476736794280046581
1097656253125000004500000000384000000002150400000008257536000002202009600000402653184000048318382080003435973836800109951162777616679880978201
114882812517187500002750000000026400000000016896000000007569408000000242221056000005536481280000088583700480000944892805120006047313952768017592186044416350277500542221
1224414062593750000001650000000001760000000000126720000000006488064000000024222105600000066437775360000013287555072000001889785610240000181419418583040010555311626649602814749767106567355827511386641
1312207031255078125000097500000000011440000000000915200000000005271552000000002249195520000000719742566400000017273821593600000307090161664000003930754069299200034304762786611200182958734861926404503599627370496154472377739119461
14610351562527343750000056875000000007280000000000064064000000000041000960000000001968046080000000071974256640000000201527918592000000429926226329600000687881962127360000800444465020928000640355572016742400315251973915934720720575940379279363243919932521508681

Generated b-files: A009965 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-3deb3ea1ada5b4a1 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-833b492313a93e6e

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,       16;
  2:      25,      160,       256;
  3:     125,     1200,      3840,       4096;
  4:     625,     8000,     38400,      81920,       65536;
  5:    3125,    50000,    320000,    1024000,     1638400,     1048576;
  6:   15625,   300000,   2400000,   10240000,    24576000,    31457280,     16777216;
  7:   78125,  1750000,  16800000,   89600000,   286720000,   550502400,    587202560,    268435456;
  8:  390625, 10000000, 112000000,  716800000,  2867200000,  7340032000,  11744051200,  10737418240,   4294967296;
  9: 1953125, 56250000, 720000000, 5376000000, 25804800000, 82575360000, 176160768000, 241591910400, 193273528320, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (5+16*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 + 16*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 + 16*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.

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-189a8da608edbb49
  • 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-189a8da608edbb49
  • 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-189a8da608edbb49

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 (1)

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