Real-rooted atlas

A093375 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 11\,x + 11\,x^{2} - x^{3} - 2\,n + 21\,n x - 24\,n x^{2} + 3\,n x^{3} + n^{2} - 13\,n^{2} x + 18\,n^{2} x^{2} - 3\,n^{2} x^{3} + 3\,n^{3} x - 6\,n^{3} x^{2} + n^{3} x^{3} + n^{4} x^{2}\right)\,P_{n}(x) = \left(4\,n x + 2\,n x^{2} - 2\,n x^{3} + n^{2} - 3\,n^{2} x + 2\,n^{2} x^{2} + 6\,n^{2} x^{3} + 3\,n^{3} x + n^{3} x^{2} - n^{3} x^{3} + n^{3} x^{4} + n^{4} x^{2} + n^{4} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
24812
39482784
41614421664440
5253208106401251920
6366002160288015002167392
7491008472589607875302434325984
86415689072224002800018144548851285248
981230415876483847875072576370449216729264960
101003240259209408018900022680016464069120145801000788480
111214400400951689604042505987525659503379201202852200013312264064
121445808594002851207920001397088162993612672006415201980003194417286309888
1316974888494245760014478752965248412011639536642606175114400031145444928219717145856
1419694641179367047042502500583783294174081076275287567485005000193793647174461516274445588480
Technical details
Node id
n-92858ff3c6477e37 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-329289f0f6a2a0c6

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:  1;
  2:  4,    8;
  3:  9,   48,    27;
  4: 16,  144,   216,    64;
  5: 25,  320,   810,   640,   125;
  6: 36,  600,  2160,  2880,  1500,   216;
  7: 49, 1008,  4725,  8960,  7875,  3024,   343;
  8: 64, 1568,  9072, 22400, 28000, 18144,  5488,  512;
  9: 81, 2304, 15876, 48384, 78750, 72576, 37044, 9216, 729;
%F Formula
The row polynomials satisfy (n^2-2*n+1+(3*n^3-13*n^2+21*n-11)*x+(n^4-6*n^3+18*n^2-24*n+11)*x^2+(n^3-3*n^2+3*n-1)*x^3)*P(n,x) = (n^2+(3*n^3-3*n^2+4*n)*x+(n^4+n^3+2*n^2+2*n)*x^2+(n^4-n^3+6*n^2-2*n)*x^3+n^3*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + (3*n^3 - 3*n^2 + 4*n)*x + (n^4 + n^3 + 2*n^2 + 2*n)*x^2 + (n^4 - n^3 + 6*n^2 - 2*n)*x^3 + n^3*x^4)*P[n - 1, x])/(n^2 - 2*n + 1 + (3*n^3 - 13*n^2 + 21*n - 11)*x + (n^4 - 6*n^3 + 18*n^2 - 24*n + 11)*x^2 + (n^3 - 3*n^2 + 3*n - 1)*x^3)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + (3*n^3 - 3*n^2 + 4*n)*x + (n^4 + n^3 + 2*n^2 + 2*n)*x^2 + (n^4 - n^3 + 6*n^2 - 2*n)*x^3 + n^3*x^4)*v[i - 1])/(n^2 - 2*n + 1 + (3*n^3 - 13*n^2 + 21*n - 11)*x + (n^4 - 6*n^3 + 18*n^2 - 24*n + 11)*x^2 + (n^3 - 3*n^2 + 3*n - 1)*x^3)); 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.