Real-rooted atlas

A013611 → D+1

nonnegative coefficients only rational roots

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

Recurrence
\(\left(-3 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(1 + 8\,x + 16\,x^{2} + 4\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
29401665
31310824064425
41720886412802562625
521340208057606400102415625
6255044080166403456030720409690625
72970070563808011648019353614336016384515625
83392811200752643046407454721032192655360655362890625
937118816704134400677376219340844728325308416294912026214416015625
10411480237602227201344000541900814622720255590402654208013107200104857687890625
11451804325603484802449920118272003973939291914240140574720129761280576716804194304478515625
124921604329652096041817602351923294617600272498688551485440749731840622854144251658240167772162587890625
1353254856160750464677248043490304203833344702873600177124147231863603203898605568294440140810905190406710886413916015625
1457296871344104832010506496758517764059095041630666752492011520011021058048178436177921984744652813740539904469762048026843545674462890625

Generated b-files: A081040 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-bf9d0f008400b3c6 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-236e6d52004bfa9f

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:  9,   40,    16;
  3: 13,  108,   240,     64;
  4: 17,  208,   864,   1280,    256;
  5: 21,  340,  2080,   5760,   6400,    1024;
  6: 25,  504,  4080,  16640,  34560,   30720,    4096;
  7: 29,  700,  7056,  38080, 116480,  193536,  143360,   16384;
  8: 33,  928, 11200,  75264, 304640,  745472, 1032192,  655360,   65536;
  9: 37, 1188, 16704, 134400, 677376, 2193408, 4472832, 5308416, 2949120, 262144;
%F Formula
The row polynomials satisfy (4*n-3+4*x)*P(n,x) = (4*n+1+(16*n+8)*x+16*x^2)*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[Together[((4*n + 1 + (16*n + 8)*x + 16*x^2)*P[n - 1, x])/(4*n - 3 + 4*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*n + 1 + (16*n + 8)*x + 16*x^2)*v[i - 1])/(4*n - 3 + 4*x)); 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-5d20fd95bc57193f
  • 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-5d20fd95bc57193f
  • 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-5d20fd95bc57193f
  • 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-5d20fd95bc57193f

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

No candidate equalities or related sequences.

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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.