Real-rooted atlas

A038337 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(23 + 68\,x + 45\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 529 + 2070\,x + 2025\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0529207020254624
1121677141513972591125314432
2279841219006064273508383500410062521381376
36436343629642252463817504820512504715718751845281251453933568
414803588917378126108500170375221743575003253845937525464881250830376562598867482624
53404825447466313050352737054860758925178893751746230653125204992294062513369062656253736694531256722988818432
6783109852811225737160920839363490630032844658329000803266100437501257286070250001229953764375006875517937500016815125390625457163239653376
718011526614633171594903880524821177508630011331407123505003325521655811250650645541354375084866809741875007116161065312500348073095585937575668064257812531087100296429568

All 15 rows

Technical details
Node id
n-0bb7630b749d3bfa · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e8b28356a6da96c8

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             529,              2070,               2025;
  1:           12167,             71415,             139725,               91125;
  2:          279841,           2190060,            6427350,             8383500,             4100625;
  3:         6436343,          62964225,          246381750,           482051250,           471571875,            184528125;
  4:       148035889,        1737812610,         8500170375,         22174357500,         32538459375,          25464881250,           8303765625;
  5:      3404825447,       46631305035,       273705486075,        892517889375,       1746230653125,        2049922940625,        1336906265625,         373669453125;
  6:     78310985281,     1225737160920,      8393634906300,      32844658329000,      80326610043750,      125728607025000,      122995376437500,       68755179375000,       16815125390625;
  7:   1801152661463,    31715949038805,    248211775086300,    1133140712350500,    3325521655811250,     6506455413543750,     8486680974187500,     7116161065312500,     3480730955859375,      756680642578125;
  8:  41426511213649,   810518697658350,   7136088533731125,   37231766262945000,  127478330139431250,   299296949023012500,   487984156015781250,   545572348340625000,   400284059923828125,   174036547792968750,   34050628916015625;
  9: 952809757913927, 20506123050756255, 200603377670441625, 1177454608065635625, 4607431075039443750, 12620354683803693750, 24691998294398531250, 34507451032544531250, 33757289053576171875, 22015623295810546875, 8614809115751953125, 1532278301220703125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (23+68*x+45*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 2070*x + 2025*x^2.
%t Mathematica
P[0, x_] := 529 + 2070*x + 2025*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((23 + 68*x + 45*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 529 + 2070*x + 2025*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((23 + 68*x + 45*x^2)*v[i - 1])/(1 + x)); 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.