Real-rooted atlas

A027466 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 2\,x + 6\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(32 + 48\,x + 24\,x^{2} + 4\,x^{3} + 48\,n x + 40\,n x^{2} + 8\,n x^{3} + 8\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181624
264256144464
35123072345610248064
440963276855296327686400131328
532768327680737280655360256000368642045952
626214431457288847360104857606144000176947220070430855168
72097152293601289909043214680064011468800049545216112394241048576453869568
81677721626843545610569646081879048192183500800010569646083596615686710886453084166545276928
9134217728241591910410871635968225485783042642411520019025362944863187763224159191043822059522621440092876046336
10107374182421474836480108716359680257698037760352321536000304405807104172637552640644245094401528823808020971520001268776961300264648704
118589934592188978561024106300440576028346784153604429185024000446461850419230384209264641417339207680448454983680922746880001116523724860397977617997313867776
12687194767361649267441664102048422952963023656976384053150220288000612290537717764861473482342427212912787456107629196083202952790016000535931387904579820584962835349504246678779068416
13549755813888142936511610889648214533734431446032554393661418032332800079597769903308872227606023372847169048831590422386872785305676772540416000185789547806723015067041792294876348416131533373443352453768544256
144398046511104123145302310912900500023148544320177786008371268788196212736009905500254633984101118648432721927547047813054464417888291992371217197049053184005202107338588161125625028935681651307551129614731737825286039797760045222461473554432
Technical details
Node id
n-158775f717def006 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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:         8,         16;
  2:        64,        256,         144;
  3:       512,       3072,        3456,        1024;
  4:      4096,      32768,       55296,       32768,        6400;
  5:     32768,     327680,      737280,      655360,      256000,       36864;
  6:    262144,    3145728,     8847360,    10485760,     6144000,     1769472,     200704;
  7:   2097152,   29360128,    99090432,   146800640,   114688000,    49545216,   11239424,    1048576;
  8:  16777216,  268435456,  1056964608,  1879048192,  1835008000,  1056964608,  359661568,   67108864,   5308416;
  9: 134217728, 2415919104, 10871635968, 22548578304, 26424115200, 19025362944, 8631877632, 2415919104, 382205952, 26214400;
%F Formula
The row polynomials satisfy (4+(6*n-2)*x+n^2*x^2)*P(n,x) = (32+(48*n+48)*x+(8*n^2+40*n+24)*x^2+(4*n^2+8*n+4)*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[((32 + (48*n + 48)*x + (8*n^2 + 40*n + 24)*x^2 + (4*n^2 + 8*n + 4)*x^3)*P[n - 1, x])/(4 + (6*n - 2)*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] = ((32 + (48*n + 48)*x + (8*n^2 + 40*n + 24)*x^2 + (4*n^2 + 8*n + 4)*x^3)*v[i - 1])/(4 + (6*n - 2)*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.