Real-rooted atlas

A131689 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 5\,x - 4\,x^{2} + n + 3\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 3\,x^{2} - 2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
325310
4624311374
52413226623375730
61208402310312020715419002
77206120213603918039842213054683133210
8504050400213360495600682206556584249271472932299754
9403204636802308320649656011305224124566128487866327071354583545375130
10362880471744027004320893289601881709202617599602404715101406961604758015170872611007179562
1136288005261760034050240012948768003204336240539148204062464747204920883260252286248276019250510224756324840104410
12399168006386688004610390400198186912005636568168011135955600015594812112015480252480010674921780648705721944132344675111622632573673895590634
1347900160083825280006678080640032023252800010293797606402336660948160384063983424046054972339203998910904824245202954229210079042954662493833903932809156759519944372341530
1462270208001183133952001031090860800545694589440019566544757760501919671052809476109586944013331839352832013977018827772010784893227108059535295547910222662741028005057242311031526858348381639455369290922
15871782912001787154969600169125884928009792301559040038751476924160011087298750528002365085750227200382104433751040046983259488765604375388396138760303787918449288015249432359561405231901766291221098209240037051064134297044322079273878443610
Technical details
Node id
n-efc52f864f8006b6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-eef4c9e321b3f244

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 -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:      1;
   2:      1,       1;
   3:      2,       5,        3;
   4:      6,      24,       31,       13;
   5:     24,     132,      266,      233,        75;
   6:    120,     840,     2310,     3120,      2071,       541;
   7:    720,    6120,    21360,    39180,     39842,     21305,      4683;
   8:   5040,   50400,   213360,   495600,    682206,    556584,    249271,     47293;
   9:  40320,  463680,  2308320,  6496560,  11305224,  12456612,   8487866,   3270713,   545835;
  10: 362880, 4717440, 27004320, 89328960, 188170920, 261759960, 240471510, 140696160, 47580151, 7087261;
%F Formula
The row polynomials satisfy P(n,x) = (n-1+(3*n-5)*x+(2*n-4)*x^2)*P(n-1,x) - (x+3*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n - 1 + (3*n - 5)*x + (2*n - 4)*x^2)*P[n - 1, x] - (x + 3*x^2 + 2*x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n - 1 + (3*n - 5)*x + (2*n - 4)*x^2)*v[i - 1] - (x + 3*x^2 + 2*x^3)*deriv(v[i - 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

Same fingerprint ?

References

  1. [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