Real-rooted atlas

A038305 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(10 + 23\,x + 13\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 100 + 260\,x + 169\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0100260169529
1100039005070219712167
210000520001014008788028561279841
31000006500001690000219700014280503712936436343
410000007800000253500004394000042841500222775804826809148035889
51000000091000000354900000768950000999635000779715300337876630627485173404825447
61000000001040000000473200000012303200000199927000002079240800013515065200501988136081573072178310985281
71000000000117000000006084000000018454800000035986860000046782918000040545195600022589466120073415764890106044993731801152661463
8100000000001300000000007605000000002636400000000599781000000093565836000001013629890000075298220400003670788244500106044993730013785849184941426511213649
910000000000014300000000009295000000000362505000000009425130000000017153736600000022299857580000020707010610000013459556896500058324746551500151644341033901792160394037952809757913927
101000000000000156000000000001115400000000004833400000000001413769500000000294064056000000044599715160000004969682546400000403786706895000023329898620600009098660462034002150592472844402329808512248121914624432020321
1110000000000000169000000000000131820000000000062834200000000002042111500000000047785409100000000828280424400000001076764551720000001049845437927000007582217051695000039427528668814000139788510734886003028751065922530302875106592253504036361936467383
12100000000000000182000000000000015379000000000000799708000000000002858956100000000007433285860000000001449490742700000000215352910344000000024496393551630000002123020774474600000137996350340849000065234638342946800021201257461457710042402514922915420393737638569928911592836324538749809
1310000000000000001950000000000000017745000000000000099963500000000000038985765000000000001114992879000000000024158179045000000000403786706895000000005249227189635000000053075519361865000000413989051022547000002446298937860505000010600628730728855000318018861921865650059060645785489335051185893014090757266635235464391245607
1410000000000000000208000000000000000202800000000000000012303200000000000000519810200000000000001621807824000000000003865308647200000000007178430344800000000001049845437927000000000121315472827120000000011039708027267920000007828156601153616000004240251492291542000001696100596916616800004724851662839146800081897428822545211206654166091831798416132610415680998648961
Technical details
Node id
n-0541c88feea53d15 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-de8557183b81eda9

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:          100,           260,           169;
  1:         1000,          3900,          5070,           2197;
  2:        10000,         52000,        101400,          87880,          28561;
  3:       100000,        650000,       1690000,        2197000,        1428050,          371293;
  4:      1000000,       7800000,      25350000,       43940000,       42841500,        22277580,         4826809;
  5:     10000000,      91000000,     354900000,      768950000,      999635000,       779715300,       337876630,        62748517;
  6:    100000000,    1040000000,    4732000000,    12303200000,    19992700000,     20792408000,     13515065200,      5019881360,       815730721;
  7:   1000000000,   11700000000,   60840000000,   184548000000,   359868600000,    467829180000,    405451956000,    225894661200,     73415764890,    10604499373;
  8:  10000000000,  130000000000,  760500000000,  2636400000000,  5997810000000,   9356583600000,  10136298900000,   7529822040000,   3670788244500,  1060449937300,   137858491849;
  9: 100000000000, 1430000000000, 9295000000000, 36250500000000, 94251300000000, 171537366000000, 222998575800000, 207070106100000, 134595568965000, 58324746551500, 15164434103390, 1792160394037;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (10+23*x+13*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 260*x + 169*x^2.
%t Mathematica
P[0, x_] := 100 + 260*x + 169*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((10 + 23*x + 13*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] = 100 + 260*x + 169*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((10 + 23*x + 13*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.