Real-rooted atlas

A085880 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-30 - 18\,x + 20\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2538
3506018128
462511256751352560
5875021000189007560113457344
613125039375047250028350085050102061376256
72062500742500011137500891000040095009622809622834603008
8335156251407656252533781252533781251520268755472967510945935938223899678208
955859375026812500005630625000675675000050675625002432430000729729000125096400938223023991418880
1094960937505127890625012306937500017229712500015506741250093040447500372161790009569874600143548119095698746652566593536
11164023437500984140625000265717968750042514875000004464061875000321412455000016070622750005509927800001239733755001652978340099178700418034567675904
122870410156250189447070312505683412109375010230141796875012276170156250010311982931250061871897587500265165275375007954958261250159099165225019091899827010413763542504967894925312
135078417968750036564609375000012066321093750002413264218750000325790669531250031275904275000002189313299250000112593255390000042222470771250011259325539000020266785970200221092210584011054610529214294475794808832
1490686035156250070735107421875002546463867187500056022205078125000840333076171875009075597222656250072604777781250000435628666687500001960329000093750065344300003125001568263200075000256624887285000256624887285001184422556700408413594137395200
151632348632812500013711728515625000053475741210937500012834177890625000002117639351953125000254116722234375000022870505001093750001568263200075000000823338180039375000329335272015750000988005816047250002155649053194000032334735797910002984744842884001279176361236011762311511156981760
Technical details
Node id
n-e0e61e54d312e320 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b1029f17d1b64336

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          1;
   2:          5,           3;
   3:         50,          60,           18;
   4:        625,        1125,          675,          135;
   5:       8750,       21000,        18900,         7560,         1134;
   6:     131250,      393750,       472500,       283500,        85050,       10206;
   7:    2062500,     7425000,     11137500,      8910000,      4009500,      962280,       96228;
   8:   33515625,   140765625,    253378125,    253378125,    152026875,    54729675,    10945935,     938223;
   9:  558593750,  2681250000,   5630625000,   6756750000,   5067562500,  2432430000,   729729000,  125096400,    9382230;
  10: 9496093750, 51278906250, 123069375000, 172297125000, 155067412500, 93040447500, 37216179000, 9569874600, 1435481190, 95698746;
%F Formula
The row polynomials satisfy n*P(n,x) = (20*n-30+(12*n-18)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((20*n - 30 + (12*n - 18)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((20*n - 30 + (12*n - 18)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.