Real-rooted atlas

A059364 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + 2\,n + 7\,n x + 5\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
2156366144
3105681147010563312
4945833427531403742217699360
510395116244519525115992012936605765763676320
613513518334171035738931184235527751004759894817873856161758080
72027025323661152213638478406295291914258960261388782019816699686434588168249662080
834459425633331980509019611423366619720670082130091229207103001408577900769218726856026381811456478480400640
96547290751361844117012584699724667811479592423480205542915417871294114833065068524482310164461364741868771424121356332697631101226041600
101374931057531937844739533373274418362065860757395083892286232235233524345445235451255100389938597714530316300519361082952840267320137209120618917296757762239288274995200
1131623414322581156656822859464372421634866203831094366630864046074826051006902797424556523456230653901734390177198191114284541731860562812035232583801168800163934374662535683465936861843456176903773724620800
12790585358062522218192479385028611487120280852232422495078887811754445215462875143999638041257803012006164727765228152406273065365791474351551387410650010836512710299725353202559023548527636000108664810698755404821142214857245081615213724540317388800
132134580466768756520698305755200919150222611523957915898886651533164647099054927042225196378361133451043766145640587219626598514421005996812687558825373289691501774020033060823350028746076831007752378843350312019823505309148793529677427173072634739200139538618057817538561414876382249517158400
1461902833536293752042557721809589253128505225282038655294820733518312662091909687546881101099618994412801868280248793176522134341194124261854549655600161427698717597168268272522632548276026744524745959164723246543276838498130500827764320728631928845104163200767886020508428165901991280524737345024990724188210504523776141487638224951715840000
Technical details
Node id
n-28cc1c51f0f8e06b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e0f3e80aae5e0ae7

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:         3,           6;
  2:        15,          63,           66;
  3:       105,         681,         1470,         1056;
  4:       945,        8334,        27531,        40374,         22176;
  5:     10395,      116244,       519525,      1159920,       1293660,        576576;
  6:    135135,     1833417,     10357389,     31184235,      52775100,      47598948,      17873856;
  7:   2027025,    32366115,    221363847,    840629529,    1914258960,    2613887820,    1981669968,     643458816;
  8:  34459425,   633331980,   5090196114,  23366619720,   67008213009,  122920710300,  140857790076,   92187268560,   26381811456;
  9: 654729075, 13618441170, 125846997246, 678114795924, 2348020554291, 5417871294114, 8330650685244, 8231016446136, 4741868771424, 1213563326976;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+1+(7*n+2)*x+(5*n+1)*x^2)*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[((2*n + 1 + (7*n + 2)*x + (5*n + 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 + (7*n + 2)*x + (5*n + 1)*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

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.