Real-rooted atlas

A038220 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 40\,x + 120\,n x + 64\,n^{2} x^{2}\right)\,P_{n}(x) = \left(125 + 600\,x + 960\,x^{2} + 512\,x^{3} + 600\,n x + 1600\,n x^{2} + 1024\,n x^{3} + 320\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
153237
225320576921
312524008640819219357
46251600086400163840102400369265
531251000007200002048000256000011796486610773
615625600000540000020480000384000003538944012845056113130121
7781253500000378000001792000004480000006193152004495769601342177281871688013
83906252000000025200000014336000004480000000825753600089915392005368709120135895449630162729441
9195312511250000016200000001075200000040320000000928972800001348730880001207959552006115295232013421772800475947501445
10976562562500000010125000000768000000003360000000009289728000001685913600000201326592000015288238080006710886400001299227607047381547294329
1148828125343750000061875000000528000000000264000000000085155840000001854504960000027682406400000280284364800001845493760000071457518387201236950581248112842477828093
12244140625187500000003712500000003520000000000198000000000007299072000000018545049600000033218887680000042042654720000036909875200000021437255516160074217034874880116135915683841704068817745489
13122070312510156250000021937500000002288000000000014300000000000059304960000000017220403200000003598712832000000546554511360000059978547200000004644738695168000241205363343360075488345194496010775213952204825464807038871733
1461035156255468750000001279687500000014560000000000010010000000000004612608000000000150678528000000003598712832000000063764692992000000839699660800000008128292716544000056281251446784000264209208180736007542649766543360989560464998400377079507707354985
Technical details
Node id
n-bbc7fa0c87e43d6a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-77290dc5d8deb0b1

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,        32;
  2:      25,       320,        576;
  3:     125,      2400,       8640,        8192;
  4:     625,     16000,      86400,      163840,      102400;
  5:    3125,    100000,     720000,     2048000,     2560000,     1179648;
  6:   15625,    600000,    5400000,    20480000,    38400000,    35389440,     12845056;
  7:   78125,   3500000,   37800000,   179200000,   448000000,   619315200,    449576960,    134217728;
  8:  390625,  20000000,  252000000,  1433600000,  4480000000,  8257536000,   8991539200,   5368709120,  1358954496;
  9: 1953125, 112500000, 1620000000, 10752000000, 40320000000, 92897280000, 134873088000, 120795955200, 61152952320, 13421772800;
%F Formula
The row polynomials satisfy (25+(120*n-40)*x+64*n^2*x^2)*P(n,x) = (125+(600*n+600)*x+(320*n^2+1600*n+960)*x^2+(512*n^2+1024*n+512)*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[((125 + (600*n + 600)*x + (320*n^2 + 1600*n + 960)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*P[n - 1, x])/(25 + (120*n - 40)*x + 64*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] = ((125 + (600*n + 600)*x + (320*n^2 + 1600*n + 960)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*v[i - 1])/(25 + (120*n - 40)*x + 64*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.