Real-rooted atlas

A038335 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(21\,n + 27\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 27\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12727
2113414582592
3357219185459049186624
4100018838578684960116212576411943936
52625493513502538026040609022320522071744535716636160
6661624362425329947010937055780140619288609039811410232452293441278242816
716209796869125047004418401936799915689034514140664426138635341704871298732224724212311581597696
8389035124856350131612370413505076477144289394495938803720786376356028703209189032123013753667282259436291848126806761930752
99190954824723945355353400084254099090300361093911194648664175807156282821018083021789090161162479972155796427033459159272686303773648836847565144260608
102144556125768702481557802675210127622972709010803828689181270324073839005638784820949358643927233408137359805090572044838513211723560144123792466254302058911320946490365203474360565760
114953924650525697636933169353303903685113336972686851263467429819206920284280171709321557043860369349438183084699325287439805330345256551730271412016646298029852371654756085151386391906114966623211075319282743446237872128
1211348991017567960416050715867703258281031831734352352318039799224039305017980102340861815355760520184213551267640368936236845994486966188632217511627590070989560139828903450759921860599266729074685379401540971589049190975618011356235639894521009714565912092213248
1325818954564967109913983495847166354100428169006347819218281412072431291922522120603492381909449015327745718432849943797438850410776492749156961582756118761348664178760299769543322660514364666945545332723457963695254021034262190521456816944917100252329691203965268321698924669147152505157427428795088896
1458390558785387156258975956482555756754598752880715114440924975635492947998252215034564114084475708667834039670264024072354345558777522452612695464592386475752581930608454475925468824561147372438244642416366400819551741603316011690206135769467110276806012722813737092464606982121548886579483852729247415318658478411730289262714112753171703869210624
Technical details
Node id
n-6678cccd35d698d6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-27210632b0078bca

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:            27;
  2:          1134,           1458;
  3:         35721,          91854,           59049;
  4:       1000188,        3857868,         4960116,          2125764;
  5:      26254935,      135025380,       260406090,        223205220,         71744535;
  6:     661624362,     4253299470,     10937055780,      14061928860,       9039811410,       2324522934;
  7:   16209796869,   125047004418,    401936799915,     689034514140,     664426138635,     341704871298,      73222472421;
  8:  389035124856,  3501316123704,  13505076477144,   28939449593880,   37207863763560,   28703209189032,   12301375366728,   2259436291848;
  9: 9190954824723, 94535535340008, 425409909030036, 1093911194648664, 1758071562828210, 1808302178909016, 1162479972155796, 427033459159272, 68630377364883;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (21*n+27*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 27.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 27;
P[n_, x_] := P[n, x] = Expand[Together[((21*n + 27*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] = 27; for(n = 2, N, my(i = n + 1); v[i] = ((21*n + 27*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.