Real-rooted atlas

A156992 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x + n\right)\,P_{n}(x) = \left(4 + 4\,x + x^{2} + 6\,n + 5\,n x + n x^{2} + 2\,n^{2} + n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
24836690
3480576216241296
4576096005760144012022680
5806401728001440005760010800720466560
6129024033868803628800201600060480090720504011022480
72322432072253440948326406773760028224000677376084672040320293932800
846448640016721510402601123840227598336012192768004064256008128512087091203628808729804160
910218700800418037760007524679680078033715200512096256002194698240060963840001045094400979776003628800285702681600
1024524881920011240570880002299207680000275904921600021459271680001126611763200402361344000958003200001437004800011975040003991680010213870867200
11637646929920032372844134400741877678080001011651379200009104862412800056652477235200247854587904007587385344000158070528000021076070400015807052800479001600395983916697600
1217854114037760099472921067520025250818424832003857763926016000394544037888000028407170727936001472964408115200552361653043200147954014208000273988915200003287866982400224172748800622702080016546470804864000
1353562342113280003249448754872320090520358171443200153188298443980800175528258633728000143614029791232000861684178747392003829707461099520012566227606732800299195895398400049865982566400054399253708800339995335680087178291200741281892057907200
141713994947624960001124809184378880000341192119261593600063364250720010240008042385668308992000737218686261657600050264910426931200002585052536242176000100529820853862400029321197749043200062831138033664000951986939904000095198693990400054922323456000130767436800035442540464018688000
Technical details
Node id
n-6bd0da8cabe63b8e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           6,           2;
  2:          48,          36,           6;
  3:         480,         576,         216,          24;
  4:        5760,        9600,        5760,        1440,         120;
  5:       80640,      172800,      144000,       57600,       10800,         720;
  6:     1290240,     3386880,     3628800,     2016000,      604800,       90720,       5040;
  7:    23224320,    72253440,    94832640,    67737600,    28224000,     6773760,     846720,      40320;
  8:   464486400,  1672151040,  2601123840,  2275983360,  1219276800,   406425600,   81285120,    8709120,   362880;
  9: 10218700800, 41803776000, 75246796800, 78033715200, 51209625600, 21946982400, 6096384000, 1045094400, 97977600, 3628800;
%F Formula
The row polynomials satisfy (n+1+x)*P(n,x) = (2*n^2+6*n+4+(n^2+5*n+4)*x+(n+1)*x^2)*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[((2*n^2 + 6*n + 4 + (n^2 + 5*n + 4)*x + (n + 1)*x^2)*P[n - 1, x])/(n + 1 + x)]];
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] = ((2*n^2 + 6*n + 4 + (n^2 + 5*n + 4)*x + (n + 1)*x^2)*v[i - 1])/(n + 1 + 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

Same fingerprint ?

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.