Real-rooted atlas

A038335 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 21\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1182139
23247564411521
358322041223814926159319
41049764898888573046667921944812313441
5188956811022480257191203000564017503290408410190224199
6340122242380855686944162401080203040945177660441082908857661213518743761
7612220032499979692817499289248340263957603969746172027788223204108065312461801088541137231006679
81101996057610285296537641998294195297996019788814291086219201333834713792778070249712259356749904378228593615352009260481
91983592903682082772548864971960522803226458925342976463031193502085402030590857642015793484448210078967422246127303216482794280046581208728361158759
103570467226624416554509772802186911176307206803723659622401389093580506240194473101270873618907107068001601260473804533440551457289483380142970408384580166798809782018140406085191601
11642684100792328247779293501444811204587875840168392160575654403929150413431936064176123419388288748721439892863366239345332440528036396181105903080141540704300734203302616433683798350277500542221317475837322472439
1211568313814261761619563933996646410392201909811814440414118538157056010608706116266227201980291808369695744269539718361430809626953971836143080961965393779718766320101909307096528624035668257483785018475659940117119736735582751138664112381557655576425121
13208229648656711683158149671293460482210704769905422336945690373792875110427582635902292190720579235353948136005129010327728082115635210512049016095801574491980428890838263776596169446514692450402782124083735231435288522129937030091121721263637664473994154472377739119461482880748567480579719
1437481336758208102461219516705073233924642480016801386905621664906745073138892869508242473776320614416218589910547808143362838253234345866425088378433764579448856678438631780134152070785923004694010434049950016175273817275319580417674358589147105276540821687921834572372324438927039190258104172324391993252150868118832349194131742609041
Technical details
Node id
n-32aabcb5e6797d69 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8e9037e0ccb64a62

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,            21;
  2:          324,           756,           441;
  3:         5832,         20412,         23814,           9261;
  4:       104976,        489888,        857304,         666792,         194481;
  5:      1889568,      11022480,      25719120,       30005640,       17503290,        4084101;
  6:     34012224,     238085568,     694416240,     1080203040,      945177660,      441082908,       85766121;
  7:    612220032,    4999796928,   17499289248,    34026395760,    39697461720,    27788223204,    10806531246,     1801088541;
  8:  11019960576,  102852965376,  419982941952,   979960197888,  1429108621920,  1333834713792,   778070249712,   259356749904,   37822859361;
  9: 198359290368, 2082772548864, 9719605228032, 26458925342976, 46303119350208, 54020305908576, 42015793484448, 21007896742224, 6127303216482, 794280046581;
%F Formula
The row polynomials satisfy P(n,x) = (18+21*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 + 21*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 + 21*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.