Real-rooted atlas

A027466 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 - 24\,x + 72\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(512 + 576\,x + 216\,x^{2} + 27\,x^{3} + 576\,n x + 360\,n x^{2} + 54\,n x^{3} + 72\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181220
26419281337
3512230419444325192
44096245763110413824202575625
5327682457604147202764808100087481059476
626214423592964976640442368019440004199043572114421385
7209715222020096557383686193152036288000117573122000376139968191972792
8167772162013265925945425927927234565806080002508226566401203289579525314412510301937
913421772818119393286115295232951268147283607552004514807808153628876832248627238263752196830032348703860
10107374182416106127360611529523201087163596801114767360007223692492830725775360859963392015305500801574640007144929411783410401
118589934592141733920768597939978240119587995648014014218240001059474898944540773646336189191946240448961356806928416000628753752255091685187484920200
1268719476736123695058124857402237911041275605286912016817061888000145299414712328652378341376363248536780810775072563202217093120003018018009624488801288981352964765749228697
1354975581388810720238370816542712067522561326629498388481943304929280001888892391260161285496210718726296307970867222412150931456576444211200010462462433281273417666569340607016312487308802296417758132
14439804651110492358976733184506531263021056135075003472281621765015207936002350621642457088179969469500620810074092753387524183601507205121291235033088002929489481318447540926218245230739928963499857849610761680259870357244787545
Technical details
Node id
n-57e795d3ee456994 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-77376c12e17ddecb

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,         12;
  2:        64,        192,         81;
  3:       512,       2304,       1944,        432;
  4:      4096,      24576,      31104,      13824,       2025;
  5:     32768,     245760,     414720,     276480,      81000,       8748;
  6:    262144,    2359296,    4976640,    4423680,    1944000,     419904,      35721;
  7:   2097152,   22020096,   55738368,   61931520,   36288000,   11757312,    2000376,    139968;
  8:  16777216,  201326592,  594542592,  792723456,  580608000,  250822656,   64012032,   8957952,   531441;
  9: 134217728, 1811939328, 6115295232, 9512681472, 8360755200, 4514807808, 1536288768, 322486272, 38263752, 1968300;
%F Formula
The row polynomials satisfy (64+(72*n-24)*x+9*n^2*x^2)*P(n,x) = (512+(576*n+576)*x+(72*n^2+360*n+216)*x^2+(27*n^2+54*n+27)*x^3)*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[((512 + (576*n + 576)*x + (72*n^2 + 360*n + 216)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(64 + (72*n - 24)*x + 9*n^2*x^2)]];
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] = ((512 + (576*n + 576)*x + (72*n^2 + 360*n + 216)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(64 + (72*n - 24)*x + 9*n^2*x^2)); 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.