Real-rooted atlas

A088729 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 7\,x + 4\,x^{2} - 4\,n - 17\,n x - 24\,n x^{2} - 13\,n x^{3} - 2\,n x^{4} + 2\,n^{2} x + 6\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(4 + 23\,x + 49\,x^{2} + 49\,x^{3} + 23\,x^{4} + 4\,x^{5} - 4\,n x - 16\,n x^{2} - 24\,n x^{3} - 16\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x) + \left(2\,x + 10\,x^{2} + 20\,x^{3} + 20\,x^{4} + 10\,x^{5} + 2\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 13 + 35\,x + 23\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
113352371
275304401173953
3541292958065019160215897
4468331726839441089716964417575316543
54729338474712734412205804211624110689202224977318943
6545835518328420597667446222015711201943295414180341833188806192579409
770872617694453335686896892810974114854770011502110525938928838332185673509884055678700945
81022475631249200538662351162620103093515386234417094883592491540714879876216141530836637513733899222457185403189015

All 15 rows

Technical details
Node id
n-3068f5c4614a7a59 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e6f2903ba541c565

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 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          13,           35,            23;
   2:          75,          304,           401,            173;
   3:         541,         2929,          5806,           5019,           1602;
   4:        4683,        31726,         83944,         108971,          69644,          17575;
   5:       47293,       384747,       1273441,        2205804,        2116241,        1068920,         222497;
   6:      545835,      5183284,      20597667,       44622201,       57112019,       43295414,       18034183,        3188806;
   7:     7087261,     76944533,     356868968,      928109741,     1485477001,     1502110525,      938928838,      332185673,       50988405;
   8:   102247563,   1249200538,    6623511626,    20103093515,    38623441709,    48835924915,    40714879876,    21614153083,     6637513733,      899222457;
   9:  1622632573,  22032243199,  131454037507,   456089759102,  1022580411854,  1551962730317,  1617784266946,  1145410405600,   527760931864,   143036657697,   17329515172;
  10: 28091567595, 419653154944, 2782597410493, 10863485753033, 27841986315406, 49308782855256, 61693515010126, 54611698087145, 33558068661996, 13645687313358, 3307212498292, 362164300173;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(4*n-3-(2*n^2-17*n+7)*x-(6*n^2-24*n+4)*x^2-(6*n^2-13*n)*x^3-(2*n^2-2*n)*x^4)*P(n-1,x) + (4-(4*n-23)*x-(16*n-49)*x^2-(24*n-49)*x^3-(16*n-23)*x^4-(4*n-4)*x^5)*d/dx P(n-1,x) + (2*x+10*x^2+20*x^3+20*x^4+10*x^5+2*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 13 + 35*x + 23*x^2.
%t Mathematica
P[1, x_] := 13 + 35*x + 23*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 3 - (2*n^2 - 17*n + 7)*x - (6*n^2 - 24*n + 4)*x^2 - (6*n^2 - 13*n)*x^3 - (2*n^2 - 2*n)*x^4)*P[n - 1, x] + (4 - (4*n - 23)*x - (16*n - 49)*x^2 - (24*n - 49)*x^3 - (16*n - 23)*x^4 - (4*n - 4)*x^5)*D[P[n - 1, x], x] + (2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 13 + 35*x + 23*x^2; for(n = 2, N, my(i = n); v[i] = (-(4*n - 3 - (2*n^2 - 17*n + 7)*x - (6*n^2 - 24*n + 4)*x^2 - (6*n^2 - 13*n)*x^3 - (2*n^2 - 2*n)*x^4)*v[i - 1] + (4 - (4*n - 23)*x - (16*n - 49)*x^2 - (24*n - 49)*x^3 - (16*n - 23)*x^4 - (4*n - 4)*x^5)*deriv(v[i - 1], x) + (2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6)*deriv(deriv(v[i - 1], x), x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.