Real-rooted atlas

A330798 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2} + 4\,n^{3} + 8\,n^{3} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2\,x - 12\,x^{2} - 18\,x^{3} - 8\,x^{4} - 7\,n - 22\,n x - 3\,n x^{2} + 32\,n x^{3} + 20\,n x^{4} + 6\,n^{2} + 26\,n^{2} x + 42\,n^{2} x^{2} + 30\,n^{2} x^{3} + 8\,n^{2} x^{4} + 13\,n^{3} + 46\,n^{3} x + 33\,n^{3} x^{2} - 20\,n^{3} x^{3} - 20\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-1 - 9\,x - 29\,x^{2} - 43\,x^{3} - 30\,x^{4} - 8\,x^{5} + 6\,n x + 30\,n x^{2} + 54\,n x^{3} + 42\,n x^{4} + 12\,n x^{5} + n^{2} + 15\,n^{2} x + 59\,n^{2} x^{2} + 97\,n^{2} x^{3} + 72\,n^{2} x^{4} + 20\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26273063
320144340264768
47070025804165249010005
5252324016380408005016624396135234
692414553939403187806008105971352447901870932
73432640645045042176720556556084436807045480249652826299968
81287027799225880401358557244040780903848401147945688256154525763058374009265
94862011934001283568079511040312989040812760480139351380015223507209622056102682438605365652250
101847565080790620353804434144002056931184647655762614028999600206575683601980151371011163787459281248187077508555135
117054322148854429376204023816469601273579956047205696384123843851208230116526640296955087660253593222960129042204732296538048241125843796944
12270415690396072136830293612413524695752404138203212282364729912047985842228634780372362570325117041643021620203206586739356148688387911731409764113016427756829900
1310400600378581840628716270063130907840427966488950206977000230073515662220001942750729920038216285207100552933511130005721797291210040116515353824170860949619503339741725404240616578338808
144011660015795911252856065940031456094670023593314244501275615503662551298762407750155959025508000360367436929500630053521972875820788886777860772847990050500497368816015890195877945917495356262861896703535648899544440
Technical details
Node id
n-b11137933d9f04e2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b24c47ab26a256b2

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:     2,       4;
  2:     6,      27,       30;
  3:    20,     144,      340,      264;
  4:    70,     700,     2580,     4165,      2490;
  5:   252,    3240,    16380,    40800,     50166,     24396;
  6:   924,   14553,    93940,   318780,    600810,    597135,     244790;
  7:  3432,   64064,   504504,  2176720,   5565560,   8443680,    7045480,    2496528;
  8: 12870,  277992,  2588040, 13585572,  44040780,  90384840,  114794568,   82561545,  25763058;
  9: 48620, 1193400, 12835680, 79511040, 312989040, 812760480, 1393513800, 1522350720, 962205610, 268243860;
%F Formula
The row polynomials satisfy (4*n^3+2*n^2+(8*n^3+4*n^2)*x+(4*n^3+2*n^2)*x^2)*P(n,x) = (13*n^3+6*n^2-7*n+(46*n^3+26*n^2-22*n-2)*x+(33*n^3+42*n^2-3*n-12)*x^2-(20*n^3-30*n^2-32*n+18)*x^3-(20*n^3-8*n^2-20*n+8)*x^4)*P(n-1,x) + (n^2-1+(15*n^2+6*n-9)*x+(59*n^2+30*n-29)*x^2+(97*n^2+54*n-43)*x^3+(72*n^2+42*n-30)*x^4+(20*n^2+12*n-8)*x^5)*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[((13*n^3 + 6*n^2 - 7*n + (46*n^3 + 26*n^2 - 22*n - 2)*x + (33*n^3 + 42*n^2 - 3*n - 12)*x^2 - (20*n^3 - 30*n^2 - 32*n + 18)*x^3 - (20*n^3 - 8*n^2 - 20*n + 8)*x^4)*P[n - 1, x] + (n^2 - 1 + (15*n^2 + 6*n - 9)*x + (59*n^2 + 30*n - 29)*x^2 + (97*n^2 + 54*n - 43)*x^3 + (72*n^2 + 42*n - 30)*x^4 + (20*n^2 + 12*n - 8)*x^5)*D[P[n - 1, x], x])/(4*n^3 + 2*n^2 + (8*n^3 + 4*n^2)*x + (4*n^3 + 2*n^2)*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] = ((13*n^3 + 6*n^2 - 7*n + (46*n^3 + 26*n^2 - 22*n - 2)*x + (33*n^3 + 42*n^2 - 3*n - 12)*x^2 - (20*n^3 - 30*n^2 - 32*n + 18)*x^3 - (20*n^3 - 8*n^2 - 20*n + 8)*x^4)*v[i - 1] + (n^2 - 1 + (15*n^2 + 6*n - 9)*x + (59*n^2 + 30*n - 29)*x^2 + (97*n^2 + 54*n - 43)*x^3 + (72*n^2 + 42*n - 30)*x^4 + (20*n^2 + 12*n - 8)*x^5)*deriv(v[i - 1], x))/(4*n^3 + 2*n^2 + (8*n^3 + 4*n^2)*x + (4*n^3 + 2*n^2)*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.