Real-rooted atlas

A062264 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 8\,n x + 4\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(16 + 48\,x + 48\,x^{2} + 16\,x^{3} + 8\,n + 28\,n x + 32\,n x^{2} + 12\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15611
215422885
335168252120575
470504126013204953649
51261260462079206435200222363
62102772138603432045045300308008134245
733055443603612012022522524024013613631824795455
84951029684084360360900900136136012252246046561259704673345
9715180181801809609603063060612612077597526046560264537049742027298155
101001300303603602333760918918023279256387987604232592029099070114406601961256158819253
1113654804868068052509602494206077597520162954792232792560223092870137287920490314007726160921406335
1218207425612252241108536062355150232792560597500904107084577613385572201144066000637408200208606320304217555334940545
13238011138421162962217072014549535064017954019632172564283383104669278610074364290005736673800292048848088223089511975985030845044155
143060162792352716042325920320089770163601438058896517681529779680029002073100401567166004015671660028231388640132334634253712555350471435600178153920965
Technical details
Node id
n-e25f0f6fc864ee61 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-71a3b30ebccc84c1

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:   5,     6;
  2:  15,    42,     28;
  3:  35,   168,    252,    120;
  4:  70,   504,   1260,   1320,     495;
  5: 126,  1260,   4620,   7920,    6435,    2002;
  6: 210,  2772,  13860,  34320,   45045,   30030,    8008;
  7: 330,  5544,  36036, 120120,  225225,  240240,  136136,   31824;
  8: 495, 10296,  84084, 360360,  900900, 1361360, 1225224,  604656,  125970;
  9: 715, 18018, 180180, 960960, 3063060, 6126120, 7759752, 6046560, 2645370, 497420;
%F Formula
The row polynomials satisfy (n^2+4*n+(2*n^2+8*n)*x+(n^2+4*n)*x^2)*P(n,x) = (n^2+8*n+16+(4*n^2+28*n+48)*x+(5*n^2+32*n+48)*x^2+(2*n^2+12*n+16)*x^3)*P(n-1,x) + ((2*n+4)*x+(6*n+12)*x^2+(6*n+12)*x^3+(2*n+4)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 8*n + 16 + (4*n^2 + 28*n + 48)*x + (5*n^2 + 32*n + 48)*x^2 + (2*n^2 + 12*n + 16)*x^3)*P[n - 1, x] + ((2*n + 4)*x + (6*n + 12)*x^2 + (6*n + 12)*x^3 + (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 8*n + 16 + (4*n^2 + 28*n + 48)*x + (5*n^2 + 32*n + 48)*x^2 + (2*n^2 + 12*n + 16)*x^3)*v[i - 1] + ((2*n + 4)*x + (6*n + 12)*x^2 + (6*n + 12)*x^3 + (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n + (2*n^2 + 8*n)*x + (n^2 + 4*n)*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

Candidate equalities ?

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.