Real-rooted atlas

A027466 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 7\,x + 21\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 147\,x + 21\,x^{2} + x^{3} + 147\,n x + 35\,n x^{2} + 2\,n x^{3} + 7\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
249569114
3343588189161136
42401548826464482511008
5168074802030870784087536104448
611764940336832413510976018375151249974848
7823543329417231765231344560300125370442401648978432
8576480126353376296475481505907242017506914886722835848181788928
940353607207532836266827932158120256529420501089093614117881128965103100738197504
1028247524916141442802334744405158120256061765725015247310424706290263424017860570001216610223104
11197732674312428910956199750354651521907464067942297501956738168380476866507091204584195269500931714458787364864
1213841287201949116836641677902979061420446966407133941237523480858016532667612485191321696268095754600039131412096169519691042816
1396889010407719746934452138808337358612926067394247213207251252670947599326924678961212920683776175207932917167150011869858550368153791964569845202944
146782230728495425784582792113360142176191151595095123270689431062252908365163704848273172747180889572864286172957073364761400290811521179786887535711920822539994735460352
Technical details
Node id
n-452b4cf2aa340550 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-18a0cc41ff95c506

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:        7,         4;
  2:       49,        56,         9;
  3:      343,       588,       189,        16;
  4:     2401,      5488,      2646,       448,       25;
  5:    16807,     48020,     30870,      7840,      875,       36;
  6:   117649,    403368,    324135,    109760,    18375,     1512,      49;
  7:   823543,   3294172,   3176523,   1344560,   300125,    37044,    2401,     64;
  8:  5764801,  26353376,  29647548,  15059072,  4201750,   691488,   67228,   3584,   81;
  9: 40353607, 207532836, 266827932, 158120256, 52942050, 10890936, 1411788, 112896, 5103, 100;
%F Formula
The row polynomials satisfy (49+(21*n-7)*x+n^2*x^2)*P(n,x) = (343+(147*n+147)*x+(7*n^2+35*n+21)*x^2+(n^2+2*n+1)*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[((343 + (147*n + 147)*x + (7*n^2 + 35*n + 21)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(49 + (21*n - 7)*x + 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] = ((343 + (147*n + 147)*x + (7*n^2 + 35*n + 21)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(49 + (21*n - 7)*x + 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.