Real-rooted atlas

A008955 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 4\,n^{2} x - n^{2} x^{2} + 8\,n^{3} + 12\,n^{3} x + 4\,n^{3} x^{2} - 8\,n^{4} - 14\,n^{4} x - 6\,n^{4} x^{2} + 4\,n^{5} + 8\,n^{5} x + 4\,n^{5} x^{2} - n^{6} - 2\,n^{6} x - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 4\,n^{2} + 2\,n^{2} x - 4\,n^{3} - 4\,n^{3} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
22321448
3320534265361155
470401771815561542857646323
52255407701681002261608253165076144002785698
69936180431438967540860967080921313556566993936518400234437598
7576943600305080584067381376967996189724543970875320764172083940244642540160026297628885
84273461400026691788040071501075568010674800048929641545317935329732866481723399403682849437209616257024003791731088277
93934757398000283797546108008954591209236016165629474318418309125631935113390668913220162480123606440175506626662562573033630976131681894400683250166093968
10440993907198000360972328529080013126154148048360278520601928295843805670792751945134815754047685101214359307474409408665378291460656215373232149177628377142387200013168189440000150453374482276668
1159127585214256000541780437811293600223253861522475112054508471155294565288739384691300248672962808257673130741074007243436386961853939625638821317980140624222979423294431362455767900089637536212322432000159335092224000039751107354989233335
129346829936415920000948135779439945000004368721526795762980001207307689065869709200222450093685340464464028717640138869971755382654419461932850292985176135484660278352510082657387387300226872026432291112358032288053486033806130817024586569299095756800022944253280256000012410857463228365602087
1317205614615445074000001914593639663384496000097579501020880331670000301254918747389946210000627663604129407840336300930235983272220567795912100645153950276988241597780213075248046331157744546863220633234730179254019676424690737613641944057016418191732811105856105623001806471319490561069083134091896832000387757880436326400004520265827545549143104526
14364896906189526237800000442026326479705607792000024696425127625043172030000842744419624354180061700001960067749143271179688311003282207522305951071098084244073278819375261502472628533796462350966824043100331372661451286799785492839888801389750197368714755137073605281855095150382825111091213997326900411924084502784239155979916977735027097622477917898315940659200076000544565519974400001899517636741553089017468426
Technical details
Node id
n-d71f2a1351c8ac58 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-78d9607790244cea

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             1;
  1:             3,              1;
  2:            23,             21,              4;
  3:           320,            534,            265,              36;
  4:          7040,          17718,          15561,            5428,             576;
  5:        225540,         770168,        1002261,          608253,          165076,           14400;
  6:       9936180,       43143896,       75408609,        67080921,        31355656,         6993936,         518400;
  7:     576943600,     3050805840,     6738137696,      7996189724,      5439708753,      2076417208,      394024464,       25401600;
  8:   42734614000,   266917880400,   715010755680,   1067480004892,    964154531793,    532973286648,   172339940368,    28494372096,    1625702400;
  9: 3934757398000, 28379754610800, 89545912092360, 161656294743184, 183091256319351, 133906689132201, 62480123606440, 17550662666256, 2573033630976, 131681894400;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(2*n^6-8*n^5+14*n^4-12*n^3+4*n^2)*x+(n^6-4*n^5+6*n^4-4*n^3+n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(2*n^4-4*n^3+2*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (2*n^6 - 8*n^5 + 14*n^4 - 12*n^3 + 4*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*v[i - 1])/(n^2 - 2*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. [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.