Real-rooted atlas

A027467 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + n x\right)\,P_{n}(x) = \left(256 + 32\,x + x^{2} + 16\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116218
2256643323
34096153614445780
4655363276846082565103173
510485766553601228801024040061837462
61677721612582912294912032768019200576732656711
7268435456234881024660602889175040716800322567848579301656
8429496729642949672961409286144234881024229376001376256501761024910258466825
9687194767367730941132828991029248563714457666060288049545216240844873728129610181369693466
10109951162777613743895347205798205849601288490188801761607680015854469129633792039321601036801600113201872665419
111759218604441624189255811072113387136614402834678415360442918502400465064427523391094784173015040608256014080019361256447829212572
122814749767106564222124650659842177033022996486047313952768010630044057600127560528691210851503308866437775362919628809011200185856230413993884992921357
134503599627370496731834939447705641165715343933441257841302175744245672129331200331657374597123224446697472230317621248121456558084685824001288601623961627041417478667116892830
1472057594037927936126100789566373888768426686420090882561422288066969655030556970188808254583545528329028450752921673701638799364534378168322099249152072161689617891328302848313615307041919020084047
Technical details
Node id
n-26842f09761592b8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4ceae639eb6d8917

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:          16,           2;
  2:         256,          64,           3;
  3:        4096,        1536,         144,          4;
  4:       65536,       32768,        4608,        256,         5;
  5:     1048576,      655360,      122880,      10240,       400,        6;
  6:    16777216,    12582912,     2949120,     327680,     19200,      576,       7;
  7:   268435456,   234881024,    66060288,    9175040,    716800,    32256,     784,     8;
  8:  4294967296,  4294967296,  1409286144,  234881024,  22937600,  1376256,   50176,  1024,    9;
  9: 68719476736, 77309411328, 28991029248, 5637144576, 660602880, 49545216, 2408448, 73728, 1296, 10;
%F Formula
The row polynomials satisfy (16+n*x)*P(n,x) = (256+(16*n+32)*x+(n+1)*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[((256 + (16*n + 32)*x + (n + 1)*x^2)*P[n - 1, x])/(16 + 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] = ((256 + (16*n + 32)*x + (n + 1)*x^2)*v[i - 1])/(16 + 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.