Real-rooted atlas

A013620 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18917
26414481289
3512172819447294913
44096184323110423328656183521
532768184320414720466560262440590491419857
62621441769472497664074649606298560283435253144124137569
72097152165150725573836810450944011757312079361856297606964782969410338673
816777216150994944594542592133772083218811699201693052928952342272306110016430467216975757441
91342177281358954496611529523216052649984270888468483047495270422856214528110199605763099363912387420489118587876497
10107374182412079595520611529523201834588569603611846246404875992432644571242905602938656153601239745564803099363912034867844012015993900449
11858993459210630044057659793997824020180474265604540606709760715145556787280453875138566465043537920363658699008013637201212803068370272883138105960934271896307633
126871947673692771293593657402237911042152583921664054487280517120980771049308161287262002216961241288359280648727808776192043639043880960147281773098243012581722464282429536481582622237229761
13549755813888804017877811254271206752256223868727853056629630797086720127500236410060819125035461509122151566489419776181538422544793611346151409049605105768134072321566542495681282937267179402425418658283299904578032905937
14439804651110469269232549888506531263021056227939068359475270518649273712641586669608658534426775049646112768344250638307164163388717220836147225415379156271104142961507754024965848425317210112164486962046534428468897277284822876792454961168377826559400929
Technical details
Node id
n-836b519045c1a1c4 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-cb7716efe3e0f2cd

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:         8,          9;
  2:        64,        144,         81;
  3:       512,       1728,       1944,         729;
  4:      4096,      18432,      31104,       23328,        6561;
  5:     32768,     184320,     414720,      466560,      262440,       59049;
  6:    262144,    1769472,    4976640,     7464960,     6298560,     2834352,      531441;
  7:   2097152,   16515072,   55738368,   104509440,   117573120,    79361856,    29760696,     4782969;
  8:  16777216,  150994944,  594542592,  1337720832,  1881169920,  1693052928,   952342272,   306110016,   43046721;
  9: 134217728, 1358954496, 6115295232, 16052649984, 27088846848, 30474952704, 22856214528, 11019960576, 3099363912, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (8+9*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[(8 + 9*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] = (8 + 9*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

Candidate equalities ?

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.