Real-rooted atlas

A178821 → 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

\(\left(n\right)\,P_{n}(x) = \left(20 + 12\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1251540
2375450135960
343757875472594517920
44375010500094500378005670286720
539375011812501417500850500255150306184128768
632812501181250017718750141750006378750153090015309055050240
725781250108281250194906250194906250116943750420997508419950721710692060160
81933593759281250001949062500233887500017541562508419950002525985004330260032476958304721920
91396484375754101562518098437500253378125002280403125013682418750547296750014073345002111001751407334595965675520
1097753906255865234375015836132812525337812500026604703125019155386250095776931250328378050007388506125985134150591080491074815565824
11666503906254398925781251319677734375237541992187528505039062502394423281250143665396875061570884375018471265312536942530625443310367524180565511725260718080
12444335937500319921875000010557421875000211148437500002850503906250027364837500000191553862500009851341500000369425306250098513415000017732414700019344452400967222620125069447659520
1329052734375002266113281250081580078125000179476171875000269214257812500290751398437500232601118750000139560671250000628023020625002093410068750050241841650008221392270008221392270037944887401308418837053440
1418676757812500156884765625000611850585937500146844140625000024229283203125002907513984375000261676258593750017943514875000009420345309375003768138123750001130441437125002466417681000036996265215003415039866001463588514013458022323978240
Technical details
Node id
n-8c7af6eedfa5a565 · 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 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:         25,         15;
  2:        375,        450,         135;
  3:       4375,       7875,        4725,         945;
  4:      43750,     105000,       94500,       37800,        5670;
  5:     393750,    1181250,     1417500,      850500,      255150,       30618;
  6:    3281250,   11812500,    17718750,    14175000,     6378750,     1530900,     153090;
  7:   25781250,  108281250,   194906250,   194906250,   116943750,    42099750,    8419950,     721710;
  8:  193359375,  928125000,  1949062500,  2338875000,  1754156250,   841995000,  252598500,   43302600,   3247695;
  9: 1396484375, 7541015625, 18098437500, 25337812500, 22804031250, 13682418750, 5472967500, 1407334500, 211100175, 14073345;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+20+(3*n+12)*x)*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[((5*n + 20 + (3*n + 12)*x)*P[n - 1, x])/(n)]];
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] = ((5*n + 20 + (3*n + 12)*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 - 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.