Real-rooted atlas

A136572 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2166480
37257611521800
43844608184322457648000
52400384002304006144006144001500000
617280345600276480011059200221184001769472054000000
71411203386880338688001806336005419008008670412805780275202205000000
8129024036126720433520640289013760011560550400277453209603699376128021139292160100800000000
913063680418037760585252864046820229120234101145600749123665920149824733184017122826649608561413324805103000000000
1014515200052254720008360755200078033715200046820229120001872809164800049941577728000856141332480008561413324800038050725888000283500000000000
11175633920070253568000126456422400013488685056000944207953920004532198178816001510732726272000345310337433600051796550615040004604137832448000184165513297920017151750000000000
122299207680010116513792002023302758400024279633100800019423706480640001087727562915840043509102516633600124311721476096000248623442952192000331497923936256000265198339149004800964357596905472001122660000000000000
13323805081600155426439168003419381661696004559175548928000410325799403520002626085116182528001225506387551846400420173618589204480010504340464730112000186743830484090880002240925965809090560016297643387702476800543254779590082560079053975000000000000
1448819843072002538631839744006092716415385600893598407589888008935984075898880006433908534647193600343141788514516992001372567154058067968004117701462174203904009150447693720453120001464071630995272499200159716905199484272640010647793679965618176003276244209220190208005959453500000000000000
Technical details
Node id
n-9b2827e5f6c3d47c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-03d7b8417365c509

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        4;
  2:       16,        64;
  3:       72,       576,       1152;
  4:      384,      4608,      18432,       24576;
  5:     2400,     38400,     230400,      614400,       614400;
  6:    17280,    345600,    2764800,    11059200,     22118400,     17694720;
  7:   141120,   3386880,   33868800,   180633600,    541900800,    867041280,     578027520;
  8:  1290240,  36126720,  433520640,  2890137600,  11560550400,  27745320960,   36993761280,   21139292160;
  9: 13063680, 418037760, 5852528640, 46820229120, 234101145600, 749123665920, 1498247331840, 1712282664960, 856141332480;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n^2+4*n^2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n^2*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 4*n^2*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.