Real-rooted atlas

A013609 → D → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 2\,x + n - 4\,n x + 2\,n^{2} x\right)\,P_{n}(x) = \left(n + 2\,n x + 2\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
241620
364872126
4896288256648
51016072012808002970
612240144038404800230412636
714336252089601680016128627251030
81644840321792044800645125017616384198288
91857660483225610080019353622579214745641472747954
10207208640537602016004838407526407372804147201024002755620
11228801188084480369600106444820697602703360228096011264002478089959598
122410561584012672063360021288964967424811008091238406758400297369658982435429400
132612482059218304010296003953664107627522108620829652480292864001932902476677121384448124357194
1428145626208256256160160069189122152550449201152830269441025024009020211253673984193822723211264431530092
Technical details
Node id
n-f9ef763bded0c707 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7cabb77b45806852

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  2;
  2:  4,  16;
  3:  6,  48,   72;
  4:  8,  96,  288,   256;
  5: 10, 160,  720,  1280,    800;
  6: 12, 240, 1440,  3840,   4800,   2304;
  7: 14, 336, 2520,  8960,  16800,  16128,   6272;
  8: 16, 448, 4032, 17920,  44800,  64512,  50176,  16384;
  9: 18, 576, 6048, 32256, 100800, 193536, 225792, 147456, 41472;
%F Formula
The row polynomials satisfy (n-1+(2*n^2-4*n+2)*x)*P(n,x) = (n+(2*n^2+2*n)*x+4*n^2*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + (2*n^2 + 2*n)*x + 4*n^2*x^2)*P[n - 1, x])/(n - 1 + (2*n^2 - 4*n + 2)*x)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n + (2*n^2 + 2*n)*x + 4*n^2*x^2)*v[i - 1])/(n - 1 + (2*n^2 - 4*n + 2)*x)); 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.