Real-rooted atlas

A027467 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + 3\,n x\right)\,P_{n}(x) = \left(256 + 96\,x + 9\,x^{2} + 48\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116622
225619227475
340964608129610810108
46553698304414726912405212629
51048576196608011059202764803240014584430914
616777216377487362654208088473601555200139968510391615663
7268435456704643072594542592247726080580608007838208571536174961881835240
842949672961288490188812683575296634178764818579456003344302083657830422394885904938436484777
968719476736231928233984260919263232152202903552535088332801203948748817557585921612431368503056196830781243899886
10109951162777641231686041605218385264640347892350976014269022208003852635996167023034368085996339206802444803149280064953915811697191171
11175921860444167256776743321610204842295296076536317214720358763986944001130106558873624721080975363783838924803990767616027713664001143188642125764318815445405652
12281474976710656126663739519795219593297206968321632774767247360861033568665600309972084719616791074591211521452994147123219155684556801773674496001097461094440814668869087336406964239402045
134503599627370496219550481834311683704914380954009633961715158745088198994424758272008059274202710016235062164245708850370463766937679687647756288922310737920076090635878442447255552143701646422320522128372265305837338
1472057594037927936378302368699121664691584017778081792691584017778081792445747511458529280200586380156338176658174059887984641611854840542003229750055162347524131952105881604261075609190431693950812161609458439684999796928717445352565231991197680599
Technical details
Node id
n-6d9acc53ce59df2b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8b59207f849e7cb7

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:          16,            6;
  2:         256,          192,           27;
  3:        4096,         4608,         1296,          108;
  4:       65536,        98304,        41472,         6912,         405;
  5:     1048576,      1966080,      1105920,       276480,       32400,        1458;
  6:    16777216,     37748736,     26542080,      8847360,     1555200,      139968,       5103;
  7:   268435456,    704643072,    594542592,    247726080,    58060800,     7838208,     571536,     17496;
  8:  4294967296,  12884901888,  12683575296,   6341787648,  1857945600,   334430208,   36578304,   2239488,   59049;
  9: 68719476736, 231928233984, 260919263232, 152202903552, 53508833280, 12039487488, 1755758592, 161243136, 8503056, 196830;
%F Formula
The row polynomials satisfy (16+3*n*x)*P(n,x) = (256+(48*n+96)*x+(9*n+9)*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[((256 + (48*n + 96)*x + (9*n + 9)*x^2)*P[n - 1, x])/(16 + 3*n*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] = ((256 + (48*n + 96)*x + (9*n + 9)*x^2)*v[i - 1])/(16 + 3*n*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.