Real-rooted atlas

A038335 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 42\,x + 126\,n x + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1029 + 2646\,x + 2268\,x^{2} + 648\,x^{3} + 2646\,n x + 3780\,n x^{2} + 1296\,n x^{3} + 756\,n^{2} x^{2} + 648\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1217293
2441302429166381
392619525618370893312381537
4194481266716877157367838208262440021039993
5408410170013160270050760411505920275562000680244481099240389
6857661211764331632850659894017283248640173604060008571080448166659897655238030757
7180108854143226124984250094008836635159387520850659894000629974412928244990049472391820820482695087048329
83782285936110374269996167002632247408213413554206723572771554800035278567123968205791641556486582589784064892616806656128479890945393
979428004658124509212865928189071070680016672252695751168135050764771440016669122966074881296487341805824622054734594048168704576457984198359290368006011129785560237
101667988097820157188163353832049631156053504202016758087253504047267767670004000700103164575144966806558544480576043543831421583360177139805280883204165545097728000432026534421504276918311146547421
11350277500542221132104657347351921273866338706607805823388976944492801559836333110132000269539718361430809631446300475500261122514656264596439040136397650066280064048112045878758400099798129451367424925465105140940812591955843624454193
12735582751138664130263976046847894432101431735406516561630548913544457984049134844492969158000970342986101150914561320744619971010967041267386757356605276168593051954175644032040414118538157056000125745643108722954242332172064955170816195504503461023744566254787692012957161
13154472377739119461688505455065789597679669916943327082008445139853397637029632149042361628673112600033112954400701774959365150904017886942771456576660974597255400652846918063669799016414722758263590229219072000114428535228937888358431834148686638081638453372729444859482112408130111367154892825221232944409177971477
14324391993252150868115570815676103241668819519129651115135091961189737426353684424652843818454318829895104400108168984375625798200576189295722657345146851008242197609330847268274176229898511982015180432128162185899105478081433600841049733932693479434243119746571290532000563278457912283943438704641199902527419435384832843330077059682304001114005886945177697411733
Technical details
Node id
n-69caac66622b328f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-950975dd37953e66

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:           21,             72;
  2:          441,           3024,            2916;
  3:         9261,          95256,          183708,           93312;
  4:       194481,        2667168,         7715736,         7838208,          2624400;
  5:      4084101,       70013160,       270050760,       411505920,        275562000,         68024448;
  6:     85766121,     1764331632,      8506598940,     17283248640,      17360406000,       8571080448,       1666598976;
  7:   1801088541,    43226124984,    250094008836,    635159387520,     850659894000,     629974412928,     244990049472,     39182082048;
  8:  37822859361,  1037426999616,   7002632247408,  21341355420672,   35727715548000,   35278567123968,   20579164155648,   6582589784064,    892616806656;
  9: 794280046581, 24509212865928, 189071070680016, 672252695751168, 1350507647714400, 1666912296607488, 1296487341805824, 622054734594048, 168704576457984, 19835929036800;
%F Formula
The row polynomials satisfy (49+(126*n-42)*x+36*n^2*x^2)*P(n,x) = (1029+(2646*n+2646)*x+(756*n^2+3780*n+2268)*x^2+(648*n^2+1296*n+648)*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[((1029 + (2646*n + 2646)*x + (756*n^2 + 3780*n + 2268)*x^2 + (648*n^2 + 1296*n + 648)*x^3)*P[n - 1, x])/(49 + (126*n - 42)*x + 36*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] = ((1029 + (2646*n + 2646)*x + (756*n^2 + 3780*n + 2268)*x^2 + (648*n^2 + 1296*n + 648)*x^3)*v[i - 1])/(49 + (126*n - 42)*x + 36*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.