Real-rooted atlas

A147716 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(196 - 14\,x + 42\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(2744 + 588\,x + 42\,x^{2} + x^{3} + 588\,n x + 70\,n x^{2} + 2\,n x^{3} + 14\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
114418
21961129317
327442352378165490
43841643904105848962593825
5537824768320246960313601750361586250
675295361290777651861608780807350030244926578125
7105413504210827008101648736215129602401000148176480264441956250
81475789056337323212818974430724818903046722800055319042689127168817301390625
92066104678453128406016341539752961011969638416941456001742549761129430445158410206100119943281250
1028925465497682644187136059769456768020239392768039530064000487913932839530064021073920714420140001211960611328125
11404956516966412727204818944102272181580803896083107840869661408000125231242752121752597128113459203667356010780001863414431907988281250
12566939123752961943791281438721718172650557447272688467968018262889568000300554982604834090727193627261222912154028952060368000156525624192169517255400390625
13793714773254144294808344351539228427947491041281323629301170176369316211264000683762585425928863589070336826923761664560665385282746744000949588642201472307581968355664160156250
1411112006825558016444480273022320644643231423536742423584667548123136723859774077440014890829638164482171579322232322315386532659218315069252481076723648004652984336143829504301428438416225134546952392578125
Technical details
Node id
n-c26e3f71719bbbdf · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8ebfa5fb6d61c270

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:          14,           4;
  2:         196,         112,           9;
  3:        2744,        2352,         378,          16;
  4:       38416,       43904,       10584,         896,         25;
  5:      537824,      768320,      246960,       31360,       1750,        36;
  6:     7529536,    12907776,     5186160,      878080,      73500,      3024,       49;
  7:   105413504,   210827008,   101648736,    21512960,    2401000,    148176,     4802,     64;
  8:  1475789056,  3373232128,  1897443072,   481890304,   67228000,   5531904,   268912,   7168,    81;
  9: 20661046784, 53128406016, 34153975296, 10119696384, 1694145600, 174254976, 11294304, 451584, 10206, 100;
%F Formula
The row polynomials satisfy (196+(42*n-14)*x+n^2*x^2)*P(n,x) = (2744+(588*n+588)*x+(14*n^2+70*n+42)*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[((2744 + (588*n + 588)*x + (14*n^2 + 70*n + 42)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(196 + (42*n - 14)*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] = ((2744 + (588*n + 588)*x + (14*n^2 + 70*n + 42)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(196 + (42*n - 14)*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

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.