Real-rooted atlas

A352880 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + 7\,n + 14\,n x + 7\,n x^{2} + 5\,n^{2} + 10\,n^{2} x + 5\,n^{2} x^{2} + n^{3} + 2\,n^{3} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(12 + 42\,x + 48\,x^{2} + 18\,x^{3} + 16\,n + 59\,n x + 70\,n x^{2} + 27\,n x^{3} + 7\,n^{2} + 27\,n^{2} x + 33\,n^{2} x^{2} + 13\,n^{2} x^{3} + n^{3} + 4\,n^{3} x + 5\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(6\,x + 18\,x^{2} + 18\,x^{3} + 6\,x^{4} + 7\,n x + 21\,n x^{2} + 21\,n x^{3} + 7\,n x^{4} + 2\,n^{2} x + 6\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
088
1183048
232128112272
3503507004201520
4727682592345615848472
598147073501617016170600647264
612825601760056320915207321622880264224
7162415837422162162378378486486324324875161480608
820064007280040768012740002329600247520014144003359208316200
924294381321329249243699696898497613477464121938966096948129329246813008
10288134402268001935360959616029611008586051207441920058605120260467204992288264051504
113381859037180037923602275416086465808216164520360274200396301620276210220110484088193154001492153104
12392250885864327037184501399362292111367019591041470771456211423396820501662721281353920465946880748843208446316088
13450331508950501247103010392525056119635020577199505258617650946551177011951403750103578832505867052750195568425029084535047883240000
14512430081327872212459522044922881285380096554320166416893566976369546777605823161344065510565120513089740802657071872081756057601131445440271832858688
Technical details
Node id
n-3a81c23f7a892ffd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5cd009bfc7bc0bb1

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:   8;
  1:  18,   30;
  2:  32,  128,    112;
  3:  50,  350,    700,    420;
  4:  72,  768,   2592,   3456,    1584;
  5:  98, 1470,   7350,  16170,   16170,    6006;
  6: 128, 2560,  17600,  56320,   91520,   73216,    22880;
  7: 162, 4158,  37422, 162162,  378378,  486486,   324324,    87516;
  8: 200, 6400,  72800, 407680, 1274000, 2329600,  2475200,  1414400,  335920;
  9: 242, 9438, 132132, 924924, 3699696, 8984976, 13477464, 12193896, 6096948, 1293292;
%F Formula
The row polynomials satisfy (n^3+5*n^2+7*n+3+(2*n^3+10*n^2+14*n+6)*x+(n^3+5*n^2+7*n+3)*x^2)*P(n,x) = (n^3+7*n^2+16*n+12+(4*n^3+27*n^2+59*n+42)*x+(5*n^3+33*n^2+70*n+48)*x^2+(2*n^3+13*n^2+27*n+18)*x^3)*P(n-1,x) + ((2*n^2+7*n+6)*x+(6*n^2+21*n+18)*x^2+(6*n^2+21*n+18)*x^3+(2*n^2+7*n+6)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8.
%t Mathematica
P[0, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 7*n^2 + 16*n + 12 + (4*n^3 + 27*n^2 + 59*n + 42)*x + (5*n^3 + 33*n^2 + 70*n + 48)*x^2 + (2*n^3 + 13*n^2 + 27*n + 18)*x^3)*P[n - 1, x] + ((2*n^2 + 7*n + 6)*x + (6*n^2 + 21*n + 18)*x^2 + (6*n^2 + 21*n + 18)*x^3 + (2*n^2 + 7*n + 6)*x^4)*D[P[n - 1, x], x])/(n^3 + 5*n^2 + 7*n + 3 + (2*n^3 + 10*n^2 + 14*n + 6)*x + (n^3 + 5*n^2 + 7*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8; for(n = 1, N, my(i = n + 1); v[i] = ((n^3 + 7*n^2 + 16*n + 12 + (4*n^3 + 27*n^2 + 59*n + 42)*x + (5*n^3 + 33*n^2 + 70*n + 48)*x^2 + (2*n^3 + 13*n^2 + 27*n + 18)*x^3)*v[i - 1] + ((2*n^2 + 7*n + 6)*x + (6*n^2 + 21*n + 18)*x^2 + (6*n^2 + 21*n + 18)*x^3 + (2*n^2 + 7*n + 6)*x^4)*deriv(v[i - 1], x))/(n^3 + 5*n^2 + 7*n + 3 + (2*n^3 + 10*n^2 + 14*n + 6)*x + (n^3 + 5*n^2 + 7*n + 3)*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.