Real-rooted atlas

A013611 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 16\,x - 16\,x^{2} + n - 28\,n x + 48\,n x^{2} + 12\,n^{2} x - 48\,n^{2} x^{2} + 16\,n^{3} x^{2}\right)\,P_{n}(x) = \left(n - 16\,n x^{2} + 12\,n^{2} x + 48\,n^{2} x^{2} + 16\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
28128136
31238417282124
41676869121638424080
52012801728081920128000228500
6241920345602457607680008847361935000
72826886048057344026880006193152561971215137500
83235849676811468807168000247726084495769633554432111700000
936460814515220643841612800074317824202309632301989888191102976788062500
104057602073603440640322560001857945606743654401509949440191102976010485760005365625000
11447040285120540672059136000408748032185450496055364812801051066368011534336000558261862435492187500
12488448380160811008010137600081749606444508119041660944384042042654720692060160006699142348828991029248229218750000
135299844942081171456016473600015182069769643425792431845539841366386278402998927360004354442526723768833802241474381742081450820312500
1456116486289921640038425625600026568622081928685158410076395929638258815795210496245760002032073179136263818366156820641344389127365868912649026171875000
Technical details
Node id
n-af0d0a00ea5542fb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6fb0656952daab5a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:  0;
  1:  4;
  2:  8,  128;
  3: 12,  384,   1728;
  4: 16,  768,   6912,   16384;
  5: 20, 1280,  17280,   81920,   128000;
  6: 24, 1920,  34560,  245760,   768000,   884736;
  7: 28, 2688,  60480,  573440,  2688000,  6193152,   5619712;
  8: 32, 3584,  96768, 1146880,  7168000, 24772608,  44957696,  33554432;
  9: 36, 4608, 145152, 2064384, 16128000, 74317824, 202309632, 301989888, 191102976;
%F Formula
The row polynomials satisfy (n-1+(12*n^2-28*n+16)*x+(16*n^3-48*n^2+48*n-16)*x^2)*P(n,x) = (n+12*n^2*x+(16*n^3+48*n^2-16*n)*x^2+64*n^3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n + 12*n^2*x + (16*n^3 + 48*n^2 - 16*n)*x^2 + 64*n^3*x^3)*P[n - 1, x])/(n - 1 + (12*n^2 - 28*n + 16)*x + (16*n^3 - 48*n^2 + 48*n - 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((n + 12*n^2*x + (16*n^3 + 48*n^2 - 16*n)*x^2 + 64*n^3*x^3)*v[i - 1])/(n - 1 + (12*n^2 - 28*n + 16)*x + (16*n^3 - 48*n^2 + 48*n - 16)*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.