Real-rooted atlas

A038221 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 6\,x + 18\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(24 + 108\,x + 162\,x^{2} + 81\,x^{3} + 108\,n x + 270\,n x^{2} + 162\,n x^{3} + 54\,n^{2} x^{2} + 81\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
163642
2364327291197
32163888131221166428890
4129631104157464279936164025633825
57776233280157464041990404920750212576413061250
64665616796161417176050388480885735007652750426040609257428125
7279936117573121190427845290790401240029000160707758410937055783061100164907081250
8167961680621568952342272507915878414880348000257132413442624893387214693280768348678440191136390625
910077696544195584734664038445712429056160707758400347128758144472480809696396718580736188286357654387420489001657677656250
106046617636279705605509980288039182082048016070775840004165545097728708721214544079343716147205648590729620232452293400042190091252129639830078125
1136279705623944605696404065221120323252176896015152445792000458209960750089355120031980813091713164288012426899605164076709256822000278454602263864518872583696522445253906250
12217678233615672832819229092695920642586017415168013637201212800047130167391436811226144038376961885206695657472223684192892952018410221637280001002436568149896325358826026112477305916652899097813212890625
1313060694016101873413324820629366198272201709358383104118189077177600045951913206650881250913192847718424507687043547136348947340913005123589993219269600026063350771897296126889942150183683722986149892542498205702352484156785474707031250
1478364164096658258978406414440556338790415403260094709769927882482918400428884523262074881313458852490104322940922445225656324885262772782071686031188608372928005473303662098432163552918380205143041563654182954867644184927899760865651472783023662252677575175048828125
Technical details
Node id
n-f2abe6c7d1ec0934 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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:        6,        36;
  2:       36,       432,        729;
  3:      216,      3888,      13122,       11664;
  4:     1296,     31104,     157464,      279936,       164025;
  5:     7776,    233280,    1574640,     4199040,      4920750,      2125764;
  6:    46656,   1679616,   14171760,    50388480,     88573500,     76527504,     26040609;
  7:   279936,  11757312,  119042784,   529079040,   1240029000,   1607077584,   1093705578,    306110016;
  8:  1679616,  80621568,  952342272,  5079158784,  14880348000,  25713241344,  26248933872,  14693280768,   3486784401;
  9: 10077696, 544195584, 7346640384, 45712429056, 160707758400, 347128758144, 472480809696, 396718580736, 188286357654, 38742048900;
%F Formula
The row polynomials satisfy (4+(18*n-6)*x+9*n^2*x^2)*P(n,x) = (24+(108*n+108)*x+(54*n^2+270*n+162)*x^2+(81*n^2+162*n+81)*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[((24 + (108*n + 108)*x + (54*n^2 + 270*n + 162)*x^2 + (81*n^2 + 162*n + 81)*x^3)*P[n - 1, x])/(4 + (18*n - 6)*x + 9*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] = ((24 + (108*n + 108)*x + (54*n^2 + 270*n + 162)*x^2 + (81*n^2 + 162*n + 81)*x^3)*v[i - 1])/(4 + (18*n - 6)*x + 9*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.