Real-rooted atlas

A038326 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 24\,x + 24\,n\right)\,P_{n}(x) = \left(529 + 1104\,x + 576\,x^{2} + 552\,n + 576\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1472471
2163322565764465
3502551175768121613824262871
4144787348244805643648259891233177614846689
540017263173744760289468800225745920779673607962624814906727
61074869281576248587212509622720138945024008126853120224545996819110297643804896337
7282748547991805780392084840488132487005388723205835691008002730622648326287287910445864714242317531295735

All 15 rows

Technical details
Node id
n-f059c41520909aa9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-55d3e09f801165f5

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:             47,              24;
  2:           1633,            2256,             576;
  3:          50255,          117576,           81216,            13824;
  4:        1447873,         4824480,         5643648,          2598912,           331776;
  5:       40017263,       173744760,       289468800,        225745920,         77967360,          7962624;
  6:     1074869281,      5762485872,     12509622720,      13894502400,       8126853120,       2245459968,       191102976;
  7:    28274854799,    180578039208,    484048813248,     700538872320,     583569100800,     273062264832,     62872879104,      4586471424;
  8:   732037471105,   5428772121408,  17335491763968,   30979124047872,   33625865871360,   22409053470720,   8737992474624,   1724513255424,   110075314176;
  9: 18716325482159, 158120093758680, 586307389112064, 1248155407005696, 1672872698585088, 1452637405642752, 806725924945920, 269629482074112, 46561857896448, 2641807540224;
%F Formula
The row polynomials satisfy (24*n-1+24*x)*P(n,x) = (552*n+529+(576*n+1104)*x+576*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[((552*n + 529 + (576*n + 1104)*x + 576*x^2)*P[n - 1, x])/(24*n - 1 + 24*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] = ((552*n + 529 + (576*n + 1104)*x + 576*x^2)*v[i - 1])/(24*n - 1 + 24*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.