Real-rooted atlas

A112486 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 2\,x + n\right)\,P_{n}(x) = \left(x + 4\,x^{2} + 5\,x^{3} + 2\,x^{4} - n - 6\,n x - 11\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} + 2\,n^{2} + 7\,n^{2} x + 8\,n^{2} x^{2} + 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-x - 5\,x^{2} - 9\,x^{3} - 7\,x^{4} - 2\,x^{5} + 2\,n x + 7\,n x^{2} + 8\,n x^{3} + 3\,n x^{4} - n^{2} x - 2\,n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 2\,x + x^{2} + 2\,x^{3} - 3\,n - 7\,n x - 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x - 4\,x^{2} - 2\,x^{3} + n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
2711321
332968015223
417883412857351053137
5116476521800418760819094554715
6874875508245336385532303975107415103951136909
7743048050323394824729526086055205460840162162013513527392535
8704016926812848635076134608100215693170203633430108153045277026752027025750424665
97362720114922512727928784249177781651143235006527160640513627114023459436005283778503445942523028527987
1084255840152969558411428719192469898720721187432326081933157626402048858011201382073092405537629597511130394275654729075782346067621
111047358080217803813121885018955049105104168162753148077680549906757400074107033800806713326980360396259495632014142148020002566537974001374931057529145298930127
1214054739840330590371968326647043985618216513950544645040819623201537031119624002534196600535602911551679436402302532296564501208075535166503886930099552564300942455753162341432251181286063981553
132025143769605331627532800594279093150723773255724950401538440329744864428184790241208084016945039227201178280998686792011792812779247500827376679919790039068268796557001144345952950200173928778773750790585358062551746667390073131
1431186669248009108555214464011338992628496648102821702387968375263548345202401199935815171431522744243549133486404570959095904000805573800667279044204940346652936695003114360982554822001336848217069365003593737509299437550518404380193752134580466768752436109451601402429
Technical details
Node id
n-cb3dfac4334b1f3e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ebd477399443f7e8

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,         1;
  2:       7,        11,         3;
  3:      32,        96,        80,         15;
  4:     178,       834,      1285,        735,        105;
  5:    1164,      7652,     18004,      18760,       8190,        945;
  6:    8748,     75508,    245336,     385532,     303975,     107415,      10395;
  7:   74304,    805032,   3394824,    7295260,    8605520,    5460840,    1621620,     135135;
  8:  704016,   9268128,  48635076,  134608100,  215693170,  203633430,  108153045,   27702675,   2027025;
  9: 7362720, 114922512, 727928784, 2491777816, 5114323500, 6527160640, 5136271140, 2345943600, 528377850, 34459425;
%F Formula
The row polynomials satisfy (n-2-2*x)*P(n,x) = -(n^3-2*n^2+n+(2*n^3-7*n^2+6*n-1)*x+(n^3-8*n^2+11*n-4)*x^2-(3*n^2-8*n+5)*x^3+(2*n-2)*x^4)*P(n-2,x) - ((n^2-2*n+1)*x+(2*n^2-7*n+5)*x^2+(n^2-8*n+9)*x^3-(3*n-7)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2-3*n-1+(2*n^2-7*n-2)*x-(5*n-1)*x^2+2*x^3)*P(n-1,x) + ((n-2)*x+(n-4)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 2*n^2 + n + (2*n^3 - 7*n^2 + 6*n - 1)*x + (n^3 - 8*n^2 + 11*n - 4)*x^2 - (3*n^2 - 8*n + 5)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] - ((n^2 - 2*n + 1)*x + (2*n^2 - 7*n + 5)*x^2 + (n^2 - 8*n + 9)*x^3 - (3*n - 7)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - 3*n - 1 + (2*n^2 - 7*n - 2)*x - (5*n - 1)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 2)*x + (n - 4)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 2 - 2*x)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 2*n^2 + n + (2*n^3 - 7*n^2 + 6*n - 1)*x + (n^3 - 8*n^2 + 11*n - 4)*x^2 - (3*n^2 - 8*n + 5)*x^3 + (2*n - 2)*x^4)*v[i - 2] - ((n^2 - 2*n + 1)*x + (2*n^2 - 7*n + 5)*x^2 + (n^2 - 8*n + 9)*x^3 - (3*n - 7)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - 3*n - 1 + (2*n^2 - 7*n - 2)*x - (5*n - 1)*x^2 + 2*x^3)*v[i - 1] + ((n - 2)*x + (n - 4)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 2 - 2*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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.