Real-rooted atlas

A013609 → basis A008288 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-1 + 2\,x\right)\,P_{n-2}(x) + \left(6 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22924457
3169210768463
49851632920208163761
5574111890912832325283230551
63346183160808923928010032128064248169
71950255654746663244124881446882876830081282015903
81136689376665652143203930080176163248076878080691225616375393
96625109246978903928385634929280191102406668480148454420352015616512133019047
1038613965159944280287417908294641952190618528812738562312768043394565143043481610241080527769
112250586811025445938205450793223863259601783743392900432192313166208750576641215616012682247680020488777241199
1213117381216520076640144104101201870716548815873777648927004313638192816641118594304231216896329154563065856167936409671298457361
137645370045411686075069949899794414278650307213564590539290034868704429434744321490140390437624491526829706248668569672908803645448192579164900087
1444560482149258405784728677960945836106583066907211209889918248339785351684524513886081818153413125438347443212052240384194882048022308044817100800786432163844704617658057
Technical details
Node id
n-9a3599832eeb87e6 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c163ccc412c10a39

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:       5,        2;
  2:      29,       24,        4;
  3:     169,      210,       76,        8;
  4:     985,     1632,      920,      208,       16;
  5:    5741,    11890,     9128,     3232,      528,      32;
  6:   33461,    83160,    80892,    39280,    10032,    1280,      64;
  7:  195025,   565474,   666324,   412488,   144688,   28768,    3008,    128;
  8: 1136689,  3766656,  5214320,  3930080,  1761632,  480768,   78080,   6912,   256;
  9: 6625109, 24697890, 39283856, 34929280, 19110240, 6668480, 1484544, 203520, 15616, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(1-2*x)*P(n-2,x) + (6+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(1 - 2*x)*P[n - 2, x] + (6 + 2*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; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 - 2*x)*v[i - 2] + (6 + 2*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

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.