Real-rooted atlas

A038284 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(14 + 34\,x + 20\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 560\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01965604001156
127441176016800800039304
2384162195204704004480001600001336336
35378243841600109760001568000011200000320000045435424
4752953664538880230496000439040000470400000268800000640000001544804416
5105413504105413504045177216001075648000015366400000131712000006272000000128000000052523350144
614757890561686616064084330803200240945152000430259200000491724800000351232000000143360000000256000000001785793904896
720661046784265642030080151795445760050598481920001084253184000015489331200000147517440000009031680000000322560000000051200000000060716992766464
828925465497641322093568002656420300800010119696384000025299240960000043370127360000051631104000000042147840000000022579200000000071680000000000102400000000002064377754059776
940495651696646363602409472045454302924800019480415539200005565833011200000111316660224000001590238003200000016226918400000000115906560000000005519360000000000157696000000000020480000000000070188843638032384
10566939123752969718956407193607636322891366400363634423398400001168824932352000002671599845376000004452666408960000005452244582400000004868075520000000003090841600000000001324646400000000003440640000000000040960000000000002386420683693101056
11793714773254144147404172175769601263464332935168006618146505850880002363623752089600000607788964823040000011576932663296000000165384752332800000001771979489280000000014063329280000000000803618816000000000031309824000000000007454720000000000008192000000000000081138303245565435904
121111200682555801622224013651116032020636584104607744001179233377406156800046327025540956160000132362930117017600000283634850250752000000463077306531840000000578846633164800000000551282507776000000000393773219840000000000204557516800000000000730562560000000000001605632000000000000016384000000000000002758702310349224820736
13155568095557812224333360204766740480033336020476674048000206365841046077440000884425033054617600000277962153245736960000066181465058508800000001215577929646080000000017365398994944000000000192948877721600000000001653847523328000000000010739269632000000000000511393792000000000000016859136000000000000003440640000000000000003276800000000000000093795878551873643905024
14217795333780937113649781790578499911680533376327626784768000355584218417856512000016509267283686195200000566032021154955264000001482464817310597120000003025438402674688000000004862311718584320000000006174364087091200000000006174364087091200000000004811192795136000000000002863805235200000000000001258815488000000000000003853516800000000000000073400320000000000000006553600000000000000003189059870763703892770816
Technical details
Node id
n-d49e8fa6522fac4f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0fbe6fa5f58efdcf

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:           196,            560,             400;
  1:          2744,          11760,           16800,             8000;
  2:         38416,         219520,          470400,           448000,           160000;
  3:        537824,        3841600,        10976000,         15680000,         11200000,           3200000;
  4:       7529536,       64538880,       230496000,        439040000,        470400000,         268800000,          64000000;
  5:     105413504,     1054135040,      4517721600,      10756480000,      15366400000,       13171200000,        6272000000,        1280000000;
  6:    1475789056,    16866160640,     84330803200,     240945152000,     430259200000,      491724800000,      351232000000,      143360000000,       25600000000;
  7:   20661046784,   265642030080,   1517954457600,    5059848192000,   10842531840000,    15489331200000,    14751744000000,     9031680000000,     3225600000000,     512000000000;
  8:  289254654976,  4132209356800,  26564203008000,  101196963840000,  252992409600000,   433701273600000,   516311040000000,   421478400000000,   225792000000000,   71680000000000,   10240000000000;
  9: 4049565169664, 63636024094720, 454543029248000, 1948041553920000, 5565833011200000, 11131666022400000, 15902380032000000, 16226918400000000, 11590656000000000, 5519360000000000, 1576960000000000, 204800000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (14+34*x+20*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 560*x + 400*x^2.
%t Mathematica
P[0, x_] := 196 + 560*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14 + 34*x + 20*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] = 196 + 560*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14 + 34*x + 20*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.