Real-rooted atlas

A038322 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 8\,n x\right)\,P_{n}(x) = \left(361 + 304\,x + 64\,x^{2} + 152\,n x + 64\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1191635
23616081921161
368591732810944204837179
4130321438976415872155648204801161297
5247609910425680131692807393280194560019660835606547
6470458812377055043753244802809446401108992002241331218350081076168025
7893871739526913867299836311689341409280491653120014904852482440560641677721632155900587

All 15 rows

Technical details
Node id
n-2d62f0f055e70d54 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-79f6ff80867e8d02

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:           19,            16;
  2:          361,           608,           192;
  3:         6859,         17328,         10944,          2048;
  4:       130321,        438976,        415872,        155648,         20480;
  5:      2476099,      10425680,      13169280,       7393280,       1945600,        196608;
  6:     47045881,     237705504,     375324480,     280944640,     110899200,      22413312,       1835008;
  7:    893871739,    5269138672,    9983631168,    9341409280,    4916531200,    1490485248,     244056064,     16777216;
  8:  16983563041,  114415582592,  252918656256,  283978842112,  186828185600,   75517919232,   18548260864,   2550136832,   150994944;
  9: 322687697779, 2445633077904, 6178441459968, 8093397000192, 6389523947520, 3228391047168, 1057250869248, 218036699136, 25820135424, 1342177280;
%F Formula
The row polynomials satisfy (19+8*n*x)*P(n,x) = (361+(152*n+304)*x+(64*n+64)*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[((361 + (152*n + 304)*x + (64*n + 64)*x^2)*P[n - 1, x])/(19 + 8*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] = ((361 + (152*n + 304)*x + (64*n + 64)*x^2)*v[i - 1])/(19 + 8*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.