Real-rooted atlas

A038323 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-7 + 27\,x + 27\,n\right)\,P_{n}(x) = \left(400 + 1080\,x + 729\,x^{2} + 540\,n + 729\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1472774
2148025387294747
34040011988010278919683282752
4102400043632006473520370040453144116092565
52480000013824000029451600029130840012488863514348907888101942
65824000004017600000111974400001590386400011797990200404639177438742048947933106463
713376000000110073600000379663200000705438720000751457574000445964029560127461340881104603532032543894817644
8302080000000288921600000011887948800000273357504000003809369088000032462967196800160547050641603933092804328282429536481133241880681769
967328000000007340544000000035103974400000096292385280000016606468368000001851353376768000131475017147040055732761865584011946769393146376255974849876905273131910690
10148480000000000181785600000000099097344000000003159357696000000064997360064000000896749291872000008331090195456000050711792328144000188098071296346003584030817943890205891132094649354764359973577139
1132460800000000004409856000000000026995161600000000098106370560000000023458230892800000003860843187801600000443890899476640000035347625543577600001882675290182346000620723635277941800106445715292933533555906056655552318094101489125536856
127045120000000000010517299200000000007143966720000000000291547745280000000007946616015360000000015200933618534400000020848553214128640000020545807347204480000014315788345148928000067776310446564456000201114457830053143203135310159537314972150094635296999121917171068796168564413
131519616000000000000247283712000000000001845786009600000000008358441062400000000002558331464832000000000557852444278272000000088925461668426240000001045406025451307520000090144729735859656000005583157454608081920000237894849667441240560064173795180316957512091707822166466462931405255515301897626746244210277361534139918
143260416000000000000057441484800000000000046736621568000000000002325690372096000000000078987268039680000000000193409858741299200000000351447039895311360000000480197493009501696000000493954347025742803200000378607864890610555200000211043351784185496576000817493210675388990288002021474548179984161628026665812906864863836861094189891315123592092320924875101247850363975
Technical details
Node id
n-43b437e46528b4ff · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f603f8b8e60b3e79

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             1;
  1:            47,             27;
  2:          1480,           2538,             729;
  3:         40400,         119880,          102789,           19683;
  4:       1024000,        4363200,         6473520,         3700404,           531441;
  5:      24800000,      138240000,       294516000,       291308400,        124888635,         14348907;
  6:     582400000,     4017600000,     11197440000,     15903864000,      11797990200,       4046391774,        387420489;
  7:   13376000000,   110073600000,    379663200000,    705438720000,     751457574000,     445964029560,     127461340881,     10460353203;
  8:  302080000000,  2889216000000,  11887948800000,  27335750400000,   38093690880000,   32462967196800,   16054705064160,   3933092804328,    282429536481;
  9: 6732800000000, 73405440000000, 351039744000000, 962923852800000, 1660646836800000, 1851353376768000, 1314750171470400, 557327618655840, 119467693931463, 7625597484987;
%F Formula
The row polynomials satisfy (27*n-7+27*x)*P(n,x) = (540*n+400+(729*n+1080)*x+729*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[((540*n + 400 + (729*n + 1080)*x + 729*x^2)*P[n - 1, x])/(27*n - 7 + 27*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] = ((540*n + 400 + (729*n + 1080)*x + 729*x^2)*v[i - 1])/(27*n - 7 + 27*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)

polytool c48629374597

Properties

Proof chains ? JSON

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.