Real-rooted atlas

A156992 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(4 + 12\,x + 9\,x^{2} + 10\,n + 21\,n x + 9\,n x^{2} + 6\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
29618054330
31056345632406488400
41344063360103680648009720255000
519584012096002851200311040014580001749609000000
63225600246758407620480011975040097977600367416003674160362250000
759351040541900800207277056042674688005029516800329204736010287648008817984016380000000
812076646401281982464058525286400149239480320230443315200217275125760118513704960317447424002380855680822150000000
9269402112003260694528001730676326400526727577600010073664921600124439390208009777380659200457124290560010713850560007142567040045360000000000
106539968512008890269696000538014597120001903743959040004345502515200006648618848256006844166461440004609336596480001885637698560003928411872000023570471232002728687500000000
11171674173440002589827530752001760273399808000710179268198400018847065194496000344163799203840004388088439848960038718427410432000228162161525760008296805873664000155565110131200084853696435200177754500000000000
124846116667392008034351316992000606019642195968002746026503700480008309097437921280001764085302204825600268447763378995200029337505569275904002265028003510272000118644323993395200038829051488747520066185883219456000330929416097280012466203750000000000
1314640373510963200264597970039603200219337790953881600011029557487966617600374832617755115520009073534402210037760016053176250063912960020938925543561625600020022847551030804480013741169887962316800064779800900393779200192733291935055872003011457686485248000138990354760857600936485550000000000000
14471348610596864000922343531190681600083348360562475008000460609361003151360000173715530435474227200047228909837144555520009527211122320539648000144478586250575216640001648940386555478016000014015993285721563136000865693702941625958400037101158697498255360001011849782659043328000145939872498900480000625456596423859200075025262812500000000000
Technical details
Node id
n-8c70341227dc569b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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:          10,            6;
  2:          96,          180,            54;
  3:        1056,         3456,          3240,           648;
  4:       13440,        63360,        103680,         64800,           9720;
  5:      195840,      1209600,       2851200,       3110400,        1458000,         174960;
  6:     3225600,     24675840,      76204800,     119750400,       97977600,       36741600,       3674160;
  7:    59351040,    541900800,    2072770560,    4267468800,     5029516800,     3292047360,    1028764800,      88179840;
  8:  1207664640,  12819824640,   58525286400,  149239480320,   230443315200,   217275125760,  118513704960,   31744742400,    2380855680;
  9: 26940211200, 326069452800, 1730676326400, 5267275776000, 10073664921600, 12443939020800, 9777380659200, 4571242905600, 1071385056000, 71425670400;
%F Formula
The row polynomials satisfy (3*n-1+3*x)*P(n,x) = (6*n^2+10*n+4+(9*n^2+21*n+12)*x+(9*n+9)*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[((6*n^2 + 10*n + 4 + (9*n^2 + 21*n + 12)*x + (9*n + 9)*x^2)*P[n - 1, x])/(3*n - 1 + 3*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] = ((6*n^2 + 10*n + 4 + (9*n^2 + 21*n + 12)*x + (9*n + 9)*x^2)*v[i - 1])/(3*n - 1 + 3*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.