Real-rooted atlas

A110608 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(12 + 24\,x + 12\,x^{2} + 14\,n + 28\,n x + 14\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(33 + 111\,x + 123\,x^{2} + 45\,x^{3} + 43\,n + 143\,n x + 157\,n x^{2} + 57\,n x^{3} + 14\,n^{2} + 46\,n^{2} x + 50\,n^{2} x^{2} + 18\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(2 + 21\,x + 51\,x^{2} + 47\,x^{3} + 15\,x^{4} + n + 12\,n x + 30\,n x^{2} + 28\,n x^{3} + 9\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 6 + 20\,x + 15\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
06201541
12010516884377
270504126013204953649
3252231079201287010010300336365
492410296450451001001201207425618564369305
5343245045240240680680111384010581485426401162803800305
612870194480122522442325928817900113954408953560392251273547139490049
7486208314026046560246901206267492010296594010983033673547100281209504686825413442773

All 15 rows

Technical details
Node id
n-b39c08934ac51671 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-1cc143b67a71a4b2

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:      6,       20,        15;
  1:     20,      105,       168,        84;
  2:     70,      504,      1260,      1320,        495;
  3:    252,     2310,      7920,     12870,      10010,       3003;
  4:    924,    10296,     45045,    100100,     120120,      74256,      18564;
  5:   3432,    45045,    240240,    680680,    1113840,    1058148,     542640,      116280;
  6:  12870,   194480,   1225224,   4232592,    8817900,   11395440,    8953560,     3922512,     735471;
  7:  48620,   831402,   6046560,  24690120,   62674920,  102965940,  109830336,    73547100,   28120950,    4686825;
  8: 184756,  3527160,  29099070, 137287920,  411863760,  823727520, 1115464350,  1012354200,  590539950,  200300100,   30045015;
  9: 705432, 14872858, 137287920, 735471000, 2549632800, 6023507490, 9921071160, 11417105700, 9013504500, 4656977325, 1419269280, 193536720;
%F Formula
The row polynomials satisfy (4*n^2+14*n+12+(8*n^2+28*n+24)*x+(4*n^2+14*n+12)*x^2)*P(n,x) = (14*n^2+43*n+33+(46*n^2+143*n+111)*x+(50*n^2+157*n+123)*x^2+(18*n^2+57*n+45)*x^3)*P(n-1,x) + (n+2+(12*n+21)*x+(30*n+51)*x^2+(28*n+47)*x^3+(9*n+15)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 20*x + 15*x^2.
%t Mathematica
P[0, x_] := 6 + 20*x + 15*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^2 + 43*n + 33 + (46*n^2 + 143*n + 111)*x + (50*n^2 + 157*n + 123)*x^2 + (18*n^2 + 57*n + 45)*x^3)*P[n - 1, x] + (n + 2 + (12*n + 21)*x + (30*n + 51)*x^2 + (28*n + 47)*x^3 + (9*n + 15)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12 + (8*n^2 + 28*n + 24)*x + (4*n^2 + 14*n + 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 6 + 20*x + 15*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^2 + 43*n + 33 + (46*n^2 + 143*n + 111)*x + (50*n^2 + 157*n + 123)*x^2 + (18*n^2 + 57*n + 45)*x^3)*v[i - 1] + (n + 2 + (12*n + 21)*x + (30*n + 51)*x^2 + (28*n + 47)*x^3 + (9*n + 15)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12 + (8*n^2 + 28*n + 24)*x + (4*n^2 + 14*n + 12)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.