Real-rooted atlas

A130534 → D²∘x (shift +1) → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(-1 - 3\,x - 3\,x^{2} - x^{3} + 3\,n + 12\,n x + 18\,n x^{2} + 12\,n x^{3} + 3\,n x^{4} - 3\,n^{2} - 15\,n^{2} x - 30\,n^{2} x^{2} - 30\,n^{2} x^{3} - 15\,n^{2} x^{4} - 3\,n^{2} x^{5} + n^{3} + 6\,n^{3} x + 15\,n^{3} x^{2} + 20\,n^{3} x^{3} + 15\,n^{3} x^{4} + 6\,n^{3} x^{5} + n^{3} x^{6}\right)\,P_{n-3}(x) + \left(-1 - 2\,x - x^{2} - x^{3} - 2\,x^{4} - x^{5} + 3\,n + 9\,n x + 6\,n x^{2} - 6\,n x^{3} - 9\,n x^{4} - 3\,n x^{5} - 3\,n^{2} - 15\,n^{2} x - 30\,n^{2} x^{2} - 30\,n^{2} x^{3} - 15\,n^{2} x^{4} - 3\,n^{2} x^{5}\right)\,P_{n-2}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4} + 3\,n + 12\,n x + 18\,n x^{2} + 12\,n x^{3} + 3\,n x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x\)
\(P_{2}(x) = 11 + 40\,x + 35\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
211403586
350255420225950
4274177141794291162412139
517641369241538618524535313132177331
6130681178044336148373768972945068921181242924172
7109584112146048312001139403015925920132164106037860117270053809164

All 15 rows

Technical details
Node id
n-f8ee6a8fe89b99a4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-40f8138fc1d472e8

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:        3,         6;
  2:       11,        40,        35;
  3:       50,       255,       420,        225;
  4:      274,      1771,      4179,       4291,       1624;
  5:     1764,     13692,     41538,      61852,      45353,      13132;
  6:    13068,    117804,    433614,     837376,     897294,     506892,     118124;
  7:   109584,   1121460,   4831200,   11394030,   15925920,   13216410,    6037860,    1172700;
  8:  1026576,  11730708,  57712908,  160114086,  274540365,  298390686,  201002208,   76800108,   12753576;
  9: 10628640, 133918488, 739173864, 2351474444, 4759953033, 6366961458, 5633856910, 3182827692, 1042487556, 150917976;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (n^3-3*n^2+3*n-1+(6*n^3-15*n^2+12*n-3)*x+(15*n^3-30*n^2+18*n-3)*x^2+(20*n^3-30*n^2+12*n-1)*x^3+(15*n^3-15*n^2+3*n)*x^4+(6*n^3-3*n^2)*x^5+n^3*x^6)*P(n-3,x) - (3*n^2-3*n+1+(15*n^2-9*n+2)*x+(30*n^2-6*n+1)*x^2+(30*n^2+6*n+1)*x^3+(15*n^2+9*n+2)*x^4+(3*n^2+3*n+1)*x^5)*P(n-2,x) + (3*n+(12*n+3)*x+(18*n+9)*x^2+(12*n+9)*x^3+(3*n+3)*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 6*x, P(2,x) = 11 + 40*x + 35*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[2, x_] := 11 + 40*x + 35*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n^2 + 3*n - 1 + (6*n^3 - 15*n^2 + 12*n - 3)*x + (15*n^3 - 30*n^2 + 18*n - 3)*x^2 + (20*n^3 - 30*n^2 + 12*n - 1)*x^3 + (15*n^3 - 15*n^2 + 3*n)*x^4 + (6*n^3 - 3*n^2)*x^5 + n^3*x^6)*P[n - 3, x] - (3*n^2 - 3*n + 1 + (15*n^2 - 9*n + 2)*x + (30*n^2 - 6*n + 1)*x^2 + (30*n^2 + 6*n + 1)*x^3 + (15*n^2 + 9*n + 2)*x^4 + (3*n^2 + 3*n + 1)*x^5)*P[n - 2, x] + (3*n + (12*n + 3)*x + (18*n + 9)*x^2 + (12*n + 9)*x^3 + (3*n + 3)*x^4)*P[n - 1, x])/(1 + 3*x + 3*x^2 + x^3)]];
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] = 3 + 6*x; v[3] = 11 + 40*x + 35*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 3*n^2 + 3*n - 1 + (6*n^3 - 15*n^2 + 12*n - 3)*x + (15*n^3 - 30*n^2 + 18*n - 3)*x^2 + (20*n^3 - 30*n^2 + 12*n - 1)*x^3 + (15*n^3 - 15*n^2 + 3*n)*x^4 + (6*n^3 - 3*n^2)*x^5 + n^3*x^6)*v[i - 3] - (3*n^2 - 3*n + 1 + (15*n^2 - 9*n + 2)*x + (30*n^2 - 6*n + 1)*x^2 + (30*n^2 + 6*n + 1)*x^3 + (15*n^2 + 9*n + 2)*x^4 + (3*n^2 + 3*n + 1)*x^5)*v[i - 2] + (3*n + (12*n + 3)*x + (18*n + 9)*x^2 + (12*n + 9)*x^3 + (3*n + 3)*x^4)*v[i - 1])/(1 + 3*x + 3*x^2 + x^3)); 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.