Real-rooted atlas

A038284 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(28 + 36\,x + 14\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 504 + 648\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
05046481152
110584272161749655296
21975687620489797764199042359296
33457440177811203429216029393280944784094371840
45808499237340352096018048012345177607936185602040733443623878656
59487215367318708992235244217604032758016038887309440199991877124285540224135291469824
615179544576136615901184526947047424112917224448014517928857601119954511872479980505088881596846084947802324992
723907782707224590862213121106588799590428455140560896457314759014404703808949862430238771820544111081202606081785233613312178120883699712

All 15 rows

Technical details
Node id
n-7e7082631edb710e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-27210632b0078bca

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            504,             648;
  1:          10584,           27216,            17496;
  2:         197568,          762048,           979776,            419904;
  3:        3457440,        17781120,         34292160,          29393280,           9447840;
  4:       58084992,       373403520,        960180480,        1234517760,         793618560,         204073344;
  5:      948721536,      7318708992,      23524421760,       40327580160,       38887309440,       19999187712,        4285540224;
  6:    15179544576,    136615901184,     526947047424,     1129172244480,     1451792885760,     1119954511872,      479980505088,       88159684608;
  7:   239077827072,   2459086221312,   11065887995904,    28455140560896,    45731475901440,    47038089498624,    30238771820544,    11108120260608,     1785233613312;
  8:  3718988421120,  43034008872960,  221317759918080,   663953279754240,  1280481325240320,  1646333132451840,  1411142684958720,   777568418242560,   249932705863680,   35704672266240;
  9: 57272421685248, 736359707381760, 4260366878423040, 14606972154593280, 32865687347834880, 50707060479516672, 54328993370910720, 39915178803118080, 19244818351503360, 5498519529000960, 706952510871552;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (14*n+28+(18*n+36)*x)*P(n-1,x) for n >= 1, with P(0,x) = 504 + 648*x.
%t Mathematica
P[0, x_] := 504 + 648*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 28 + (18*n + 36)*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] = 504 + 648*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 28 + (18*n + 36)*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

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.