Real-rooted atlas

A013622 → basis A013611 → D

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(8\,n + 20\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 20\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12020
23208001120
33840192002400047040
4409603072007680006400001756160
5409600409600015360000256000001600000061465600
63932160491520002457600006144000007680000003840000002065244160
7367001605505024003440640000114688000002150400000021504000000896000000067464642560
83355443205872025600440401920001835008000004587520000006881280000005734400000002048000000002158868561920
930198988806039797760052848230400026424115200008257536000000165150720000002064384000000014745600000000460800000000068004359700480
10268435456006039797760006039797760000352321536000001321205760000003303014400000005505024000000005898240000000003686400000000001024000000000002115691190681600
11236223201280590558003200066437775360000442918502400000193776844800000058133053440000001211105280000000017301504000000000162201600000000009011200000000000225280000000000065163288672993280
1220615843020805669356830720070866960384000053150220288000002657511014400000093012885504000000232532213760000000415236096000000000519045120000000000432537600000000000216268800000000000491520000000000001990442272193249280
13178670639513605360119185408007370163879936000614180323328000003454764318720000001381905727488000000403055837184000000086369107968000000001349517312000000000014994636800000000000112459776000000000005111808000000000000106496000000000000060376748923195228160
1415393162788864050027779063808007504166859571200068788196212736000042992622632960000001934668018483200000064488933949440000000161222334873600000000302291877888000000000419849830400000000000419849830400000000000286261248000000000000119275520000000000000229376000000000000001820591198299425341440
Technical details
Node id
n-4fb8d1ebc18dc0c9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9629d1e9c04dbfbe

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          0;
  1:         20;
  2:        320,         800;
  3:       3840,       19200,        24000;
  4:      40960,      307200,       768000,        640000;
  5:     409600,     4096000,     15360000,      25600000,      16000000;
  6:    3932160,    49152000,    245760000,     614400000,     768000000,      384000000;
  7:   36700160,   550502400,   3440640000,   11468800000,   21504000000,    21504000000,     8960000000;
  8:  335544320,  5872025600,  44040192000,  183500800000,  458752000000,   688128000000,   573440000000,   204800000000;
  9: 3019898880, 60397977600, 528482304000, 2642411520000, 8257536000000, 16515072000000, 20643840000000, 14745600000000, 4608000000000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (8*n+20*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 20.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 20;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 20*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 20; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 20*n*x)*v[i - 1])/(n - 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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [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
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.