Real-rooted atlas

A088996 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 18\,n x - 9\,n x^{2} + 12\,n^{2} + 30\,n^{2} x + 18\,n^{2} x^{2} - 4\,n^{3} - 12\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 6\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2153651
31024504861038
4894559811502777625770
5962177202229473300024145800762120
61228411195578460865788076168354340314928026238312
718152282070183697573248243541404339806340251470008771573601032065424
8304716603976584842208864816677546499612403337220135584727128198520624211631616045689106672
9572747895840566386853655870474194702038308439528676355632362162032566466741036288960296832642831033602248937300160
10119159927551940724871921397716413066584557354681215652371287295278379134273883289554375080424911962475328109737889819202142770112000121863628474560
112718419073304861977752862389586327903901842565666579565698297058391901204430084952030176256608572132217637831477264961155306175516872447370435022400777825550656007209417208952448
126747339049290131398105643586115859535283351861079154070717082139762873390939052318535246149698911250835395756661130840170673866569800525951981114456501767026190080195048679933958403054733071667200462396548402683776
1318101660781802538112450869342703665167300098773121294484352413563683270159444248051523093724059007019024658620442189202909688867557873432465674116927368268681325659438754846568240291154750525845043290621180141979392012906247227793920031960912070116205568
14521949426147292511810901240393939012297036115035277477802004463206975108336625473349640947951043212153192199693102389419998282428717414096080218509085053445256056249895222795604986838407661111886403400778531158896143521578695316065398010884471732371610498176058375948999560192002368480590008835240960
Technical details
Node id
n-8d66c0372bb5e2eb · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-cd7a1aa301377704

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         0;
  1:         3;
  2:        15,         36;
  3:       102,        450,         486;
  4:       894,       5598,       11502,         7776;
  5:      9621,      77202,      229473,       300024,       145800;
  6:    122841,    1195578,     4608657,      8807616,      8354340,      3149280;
  7:   1815228,   20701836,    97573248,    243541404,    339806340,    251470008,     77157360;
  8:  30471660,  397658484,  2208864816,   6775464996,  12403337220,  13558472712,   8198520624,   2116316160;
  9: 572747895, 8405663868, 53655870474, 194702038308, 439528676355, 632362162032, 566466741036, 288960296832, 64283103360;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(12*n^3-30*n^2+18*n)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (12*n^3 - 30*n^2 + 18*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (12*n^3 - 30*n^2 + 18*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(n - 1)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.