Real-rooted atlas

A013610 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 12\,x - 9\,x^{2} + n - 21\,n x + 27\,n x^{2} + 9\,n^{2} x - 27\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n}(x) = \left(n - 9\,n x^{2} + 9\,n^{2} x + 27\,n^{2} x^{2} + 9\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
267278
39216729954
412432291651848544
5157207290259203037564320
61810801458077760182250157464433152
72115122551518144063787511022487501412698752
824201640824362880170100044089926001128335923215876096
92725926123665318438272501322697627005076302330881434890789358336
103032408748010886407654500330674409001692015116544014348907059049000485621760
11333960120285171072014033250727483682475465305542732807891898856495390002357826572564947968
12364752160380256608024057000145496736594111672166281984031567595403897234000282939188491833004813230931968
133956162084943706560390926252702082241287241956432333158410259468505168880140001839104724611938290624350272763166903343104
14426552265356518918460810750472864392257448391210087773696287265118145910804900085824887148835680343684903818683413124466936332591529984
Technical details
Node id
n-97674feacf20fae9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-10c18c8945c6e24a

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:  3;
  2:  6,   72;
  3:  9,  216,   729;
  4: 12,  432,  2916,   5184;
  5: 15,  720,  7290,  25920,   30375;
  6: 18, 1080, 14580,  77760,  182250,   157464;
  7: 21, 1512, 25515, 181440,  637875,  1102248,   750141;
  8: 24, 2016, 40824, 362880, 1701000,  4408992,  6001128,  3359232;
  9: 27, 2592, 61236, 653184, 3827250, 13226976, 27005076, 30233088, 14348907;
%F Formula
The row polynomials satisfy (n-1+(9*n^2-21*n+12)*x+(9*n^3-27*n^2+27*n-9)*x^2)*P(n,x) = (n+9*n^2*x+(9*n^3+27*n^2-9*n)*x^2+27*n^3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n + 9*n^2*x + (9*n^3 + 27*n^2 - 9*n)*x^2 + 27*n^3*x^3)*P[n - 1, x])/(n - 1 + (9*n^2 - 21*n + 12)*x + (9*n^3 - 27*n^2 + 27*n - 9)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n + 9*n^2*x + (9*n^3 + 27*n^2 - 9*n)*x^2 + 27*n^3*x^3)*v[i - 1])/(n - 1 + (9*n^2 - 21*n + 12)*x + (9*n^3 - 27*n^2 + 27*n - 9)*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.