Real-rooted atlas

A178822 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(25 + 15\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1301848
25256301891344
37000126007560151228672
4787501890001701006804010206516096
5787500236250028350001701000510300612368257536
67218750259875003898125031185000140332503367980336798121110528
7618750002598750004677750004677750002806650001010394002020788017321041660944384
850273437524131250005067562500608107500045608062502189187000656756100112586760844400721592276992
93910156250211148437505067562500070945875000638512875003831077250015324309000394053660059108049039405366268703891456
1029326171875175957031250475083984375760134375000798141093750574661587500287330793750985134150002216551837529554024501773241473224446697472
1121328125000014076562500004222968750000760134375000091216125000007662154500000459729270000019702683000005910804900001182160980001418593176077377809637520834297856
12151074218750010877343750000358952343750007179046875000096917132812500930404475000006512831325000033494561100000125604604125003349456110000602902099800657711381603288556908425236122042368
131045898437500081580078125000293688281250000646114218750000969171328125000104670503437500083736402750000050241841650000022608828742500075362762475000180870629940002959701217200295970121720136601594644710307813392384
1470971679687500596162109375000232503222656250055800773437500009207127617187500110485531406250009943697826562500681853565250000035797312175625001431892487025000429567746107500937238718780001405858078170012977151490805561636353251140484831117312
Technical details
Node id
n-09bd967d44af22a1 · 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:         30,          18;
  2:        525,         630,         189;
  3:       7000,       12600,        7560,        1512;
  4:      78750,      189000,      170100,       68040,       10206;
  5:     787500,     2362500,     2835000,     1701000,      510300,       61236;
  6:    7218750,    25987500,    38981250,    31185000,    14033250,     3367980,      336798;
  7:   61875000,   259875000,   467775000,   467775000,   280665000,   101039400,    20207880,    1732104;
  8:  502734375,  2413125000,  5067562500,  6081075000,  4560806250,  2189187000,   656756100,  112586760,   8444007;
  9: 3910156250, 21114843750, 50675625000, 70945875000, 63851287500, 38310772500, 15324309000, 3940536600, 591080490, 39405366;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+25+(3*n+15)*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 + 25 + (3*n + 15)*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 + 25 + (3*n + 15)*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.