Real-rooted atlas

A013610 → basis A008277

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
21633958
36427918927559
425621452727918816127
51024157833370520115405024374920
64096113553384399362610125145167677291007299
71638480715941846495867127309015069400866339218714728003
86553656992654426796788826598670437812245660235414822536926561232161484
9262144400914634597774651287549675133918029062780678114528241016966746944784196833917881441
101048576281426673471804903918117478170252895336651594586970950796333638600158807735419034445255904970373912839
114194304197313243948024770409249681363327458984973150378840125466159589543113367392970894745964960338842845123412411771471338934725490
121677721613824509985486167101407338993203447880928036603818574917303304464749032488413972510565642431049101472474202307514361307294357816253144126871729941773
1367108864968219015434903144544025455276177524351396546546695301872018397439191279695390573604887374738875210998690653808149631744576612305451401159207841811519921261594323566851569150643
1426843545667795463739349321911144879606568464413730237105732796978539757989191430513376935378493677160575230189088045458458047288079381912780170867280581367258823255673023842037583524532267478296912529364429105122

Computed in your browser ?

Technical details
Node id
n-ea153c5d201f990d · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b4cb782974a54903

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): no

OEIS matches ?

No OEIS matches.

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,        3;
  2:     16,       33,         9;
  3:     64,      279,       189,         27;
  4:    256,     2145,      2727,        918,         81;
  5:   1024,    15783,     33705,      20115,       4050,       243;
  6:   4096,   113553,    384399,     362610,     125145,     16767,       729;
  7:  16384,   807159,   4184649,    5867127,    3090150,    694008,     66339,     2187;
  8:  65536,  5699265,  44267967,   88826598,   67043781,  22456602,   3541482,   253692,   6561;
  9: 262144, 40091463, 459777465, 1287549675, 1339180290, 627806781, 145282410, 16966746, 944784, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (4+3*x)*P(n-1,x) + 3*x*d/dx 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 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], 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 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], 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-240eb6cab1dce1b5
  • 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-240eb6cab1dce1b5
  • 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-240eb6cab1dce1b5

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741