Real-rooted atlas

A125092 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 4\,x + 6\,x^{2} + 4\,x^{3} + x^{4}\right)\,P_{n}(x) = \left(-1 - 12\,x - 62\,x^{2} - 180\,x^{3} - 321\,x^{4} - 360\,x^{5} - 248\,x^{6} - 96\,x^{7} - 16\,x^{8}\right)\,P_{n-4}(x) + \left(4 + 40\,x + 168\,x^{2} + 384\,x^{3} + 516\,x^{4} + 408\,x^{5} + 176\,x^{6} + 32\,x^{7}\right)\,P_{n-3}(x) + \left(-6 - 48\,x - 156\,x^{2} - 264\,x^{3} - 246\,x^{4} - 120\,x^{5} - 24\,x^{6}\right)\,P_{n-2}(x) + \left(4 + 24\,x + 56\,x^{2} + 64\,x^{3} + 36\,x^{4} + 8\,x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 9\,x\)
\(P_{2}(x) = 1 + 18\,x + 44\,x^{2}\)
\(P_{3}(x) = 1 + 27\,x + 132\,x^{2} + 170\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11910
21184463
3127132170330
41362646805761557
51454401700288017926858
61546603400864010752524828755
7163924595020160376323673614688116154
8172123295204032010035214694411750439680455625
9181158414280725762257924408325287683571201041921745226
1019019802040012096045158411020801762560178560010419202672646554439
111992420280501900808279042424576484704065472005730560293990467225624210090
12110829043740028512014192644849152116328961964160022922240176394248067072166297688160157
131117343248620411840230630490055682520460851068160744972807643750452435968216186884055040317093130
141126400461880576576358758415759744504092161191590402085923842675312642447011841513308165677056097648641128249243
Technical details
Node id
n-32944be6042d9412 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d308d9d401341a72

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: 1,  9;
  2: 1, 18,   44;
  3: 1, 27,  132,   170;
  4: 1, 36,  264,   680,   576;
  5: 1, 45,  440,  1700,  2880,   1792;
  6: 1, 54,  660,  3400,  8640,  10752,   5248;
  7: 1, 63,  924,  5950, 20160,  37632,  36736,  14688;
  8: 1, 72, 1232,  9520, 40320, 100352, 146944, 117504,  39680;
  9: 1, 81, 1584, 14280, 72576, 225792, 440832, 528768, 357120, 104192;
%F Formula
The row polynomials satisfy (1+4*x+6*x^2+4*x^3+x^4)*P(n,x) = -(1+12*x+62*x^2+180*x^3+321*x^4+360*x^5+248*x^6+96*x^7+16*x^8)*P(n-4,x) + (4+40*x+168*x^2+384*x^3+516*x^4+408*x^5+176*x^6+32*x^7)*P(n-3,x) - (6+48*x+156*x^2+264*x^3+246*x^4+120*x^5+24*x^6)*P(n-2,x) + (4+24*x+56*x^2+64*x^3+36*x^4+8*x^5)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 9*x, P(2,x) = 1 + 18*x + 44*x^2, P(3,x) = 1 + 27*x + 132*x^2 + 170*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 9*x;
P[2, x_] := 1 + 18*x + 44*x^2;
P[3, x_] := 1 + 27*x + 132*x^2 + 170*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + 12*x + 62*x^2 + 180*x^3 + 321*x^4 + 360*x^5 + 248*x^6 + 96*x^7 + 16*x^8)*P[n - 4, x] + (4 + 40*x + 168*x^2 + 384*x^3 + 516*x^4 + 408*x^5 + 176*x^6 + 32*x^7)*P[n - 3, x] - (6 + 48*x + 156*x^2 + 264*x^3 + 246*x^4 + 120*x^5 + 24*x^6)*P[n - 2, x] + (4 + 24*x + 56*x^2 + 64*x^3 + 36*x^4 + 8*x^5)*P[n - 1, x])/(1 + 4*x + 6*x^2 + 4*x^3 + x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 9*x; v[3] = 1 + 18*x + 44*x^2; v[4] = 1 + 27*x + 132*x^2 + 170*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(1 + 12*x + 62*x^2 + 180*x^3 + 321*x^4 + 360*x^5 + 248*x^6 + 96*x^7 + 16*x^8)*v[i - 4] + (4 + 40*x + 168*x^2 + 384*x^3 + 516*x^4 + 408*x^5 + 176*x^6 + 32*x^7)*v[i - 3] - (6 + 48*x + 156*x^2 + 264*x^3 + 246*x^4 + 120*x^5 + 24*x^6)*v[i - 2] + (4 + 24*x + 56*x^2 + 64*x^3 + 36*x^4 + 8*x^5)*v[i - 1])/(1 + 4*x + 6*x^2 + 4*x^3 + x^4)); 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

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.