Real-rooted atlas

A038221 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1182442
23248645761764
3583223328311041382474088
410497655987211197449953283317763111696
51889568125971203359232044789760298598407962624130691232
634012224272097792906992640161243136016124313608599633921911029765489031744
7612220032571405363222856214528507915878406772211712054177693696240789749764586471424230539333248
81101996057611754624614454854914867214627977297922437996216320260052929740817336861982726604518850561100753141769682651996416
919835929036823803114844161269499458355239495538704384789910774087681053214365450249361905470668853496602689536178322008965122641807540224406671383849472
103570467226624476062296883202856373781299201015599566684160236973232226304037915717156208644212857461800960320979616137216016048980806860804755253572403206340338096537617080198121677824
11642684100792329426033478287366284022318858240251360892754329606702957140115456012512186661548851216682915548731801615888490998792192010592327332528128047077010366791680125538694311444481521681143169024717368321110468608
1211568313814261761850930210281881613573488208733798460326614261039104018097984278311731203860903312706502656600584959754344857668638281114782269445719856759565189120338954474640900096013558178985636003843286831269245091843652034743605657630129469486639681536
1320822964865671168360931391004966912288745112803973529614116427737083150336470547591236105011201129314218966652026882007669722607381381122676892963476508508162676892963476508508161982883676649265561601057537960879608299523845592585016757452885457613000372387848764883384653578241265437718438866624512
1437481336758208102469965161948655124486063647368883444121632339452634045035315211857799299149846282243162079813106625675264632415962621325135052896368146685154306293761124295044660133573427299937337303122984304646662489153541532286976323029777141407626035210767659238046920867842208750612932701716482103572012316858777653148384174432398229504
Technical details
Node id
n-d6888b333aa0e28a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-c91d5d506c107735

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:           18,            24;
  2:          324,           864,            576;
  3:         5832,         23328,          31104,          13824;
  4:       104976,        559872,        1119744,         995328,         331776;
  5:      1889568,      12597120,       33592320,       44789760,       29859840,         7962624;
  6:     34012224,     272097792,      906992640,     1612431360,     1612431360,       859963392,      191102976;
  7:    612220032,    5714053632,    22856214528,    50791587840,    67722117120,     54177693696,    24078974976,     4586471424;
  8:  11019960576,  117546246144,   548549148672,  1462797729792,  2437996216320,   2600529297408,  1733686198272,   660451885056,   110075314176;
  9: 198359290368, 2380311484416, 12694994583552, 39495538704384, 78991077408768, 105321436545024, 93619054706688, 53496602689536, 17832200896512, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (18+24*x)*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[(18 + 24*x)*P[n - 1, 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] = (18 + 24*x)*v[i - 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.