Real-rooted atlas

A075500 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 6\,x - 4\,x^{2} - 5\,x^{3} - 5\,n x - 10\,n x^{2} - 5\,n x^{3}\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 25 + 65\,x + 41\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0256541131
11255507553311761
26254375100009425317627601
331253500012000018287513020035451506651
415625290625140687530775003466625196898044798110674211
578125253125016565625486568757699312568372900322065656282416251686881
639062523046875197656250748109375155934437519249738751408163800565971155965462316524202561
7195312521718750023960937501138835937530061484375484025481254889842750030376496875106379039751611270851183991725451

All 15 rows

Technical details
Node id
n-ab1ac7b8d83baeb5 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d62f43983c4e064b

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:       25,          65,           41;
  1:      125,         550,          755,           331;
  2:      625,        4375,        10000,          9425,           3176;
  3:     3125,       35000,       120000,        182875,         130200,          35451;
  4:    15625,      290625,      1406875,       3077500,        3466625,        1968980,         447981;
  5:    78125,     2531250,     16565625,      48656875,       76993125,       68372900,       32206565,        6282416;
  6:   390625,    23046875,    197656250,     748109375,     1559344375,     1924973875,     1408163800,      565971155,       96546231;
  7:  1953125,   217187500,   2396093750,   11388359375,    30061484375,    48402548125,    48898427500,    30376496875,    10637903975,     1611270851;
  8:  9765625,  2095703125,  29555078125,  173553906250,   565178593750,  1140487146875,  1492335882500,  1273875890000,   686900669500,   212952636775,   28985293526;
  9: 48828125, 20527343750, 371583984375, 2667449609375, 10517507031250, 25879513312500, 42119320956250, 46320231048125, 34184585617500, 16267917590750, 4521005740580, 558413253581;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (5-(5*n-6)*x-(10*n+4)*x^2-(5*n+5)*x^3)*P(n-1,x) + (5*x+15*x^2+15*x^3+5*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 25 + 65*x + 41*x^2.
%t Mathematica
P[0, x_] := 25 + 65*x + 41*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((5 - (5*n - 6)*x - (10*n + 4)*x^2 - (5*n + 5)*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 25 + 65*x + 41*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((5 - (5*n - 6)*x - (10*n + 4)*x^2 - (5*n + 5)*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*deriv(v[i - 1], x))/(1 + x)); 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.