Real-rooted atlas

A038279 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 - 8\,x + 24\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(512 + 192\,x + 24\,x^{2} + x^{3} + 192\,n x + 40\,n x^{2} + 2\,n x^{3} + 8\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
264649137
3512768216161512
44096819234565122516281
532768819204608010240100036172044
6262144786432552960163840240001728491791153
720971527340032619315222937604480004838427446418423288
8167772166710886466060288293601287168000103219287808409681187598673
913421772860397977667947724835232153610321920018579456210739214745658321001894055724
1010737418245368709120679477248040265318401376256000297271296421478403932160233280800012118983603961
11858993459247244640256664377753604429185024017301504000435997900874180198486507520684288035200010648144189061198632
12687194767364123168604166378026434564724464025602076180480005979399782411868831744166094438416422912011264000511104138241691872403223337
135497558138883573412790272603013408358449134425866242399141888000777321971712176336928768287897026563415965696292864000177182727188481757619618452063050092
14439804651110430786325577728562812514467845002777906380826870389145600967334009241624687170027524606352424966376469299265601536004961116162683699298425621952225181037332884321
Technical details
Node id
n-e2fc71959f27e6f9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5484980da109477c

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,         4;
  2:        64,        64,         9;
  3:       512,       768,       216,        16;
  4:      4096,      8192,      3456,       512,        25;
  5:     32768,     81920,     46080,     10240,      1000,       36;
  6:    262144,    786432,    552960,    163840,     24000,     1728,      49;
  7:   2097152,   7340032,   6193152,   2293760,    448000,    48384,    2744,     64;
  8:  16777216,  67108864,  66060288,  29360128,   7168000,  1032192,   87808,   4096,   81;
  9: 134217728, 603979776, 679477248, 352321536, 103219200, 18579456, 2107392, 147456, 5832, 100;
%F Formula
The row polynomials satisfy (64+(24*n-8)*x+n^2*x^2)*P(n,x) = (512+(192*n+192)*x+(8*n^2+40*n+24)*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[((512 + (192*n + 192)*x + (8*n^2 + 40*n + 24)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(64 + (24*n - 8)*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] = ((512 + (192*n + 192)*x + (8*n^2 + 40*n + 24)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(64 + (24*n - 8)*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.