Real-rooted atlas

A038301 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(9 + 29\,x + 20\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 81 + 360\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
081360400841
1729486010800800024389
2656158320194400288000160000707281
359049656100291600064800007200000320000020511149
453144170858803936600011664000019440000017280000064000000594823321
54782969744017404960116001837080000408240000054432000004032000000128000000017249876309
643046721765275040595213920026453952000734832000001306368000001451520000009216000000025600000000500246412961
73874204897748409780688747536003571283520001190427840000264539520000039191040000003732480000000207360000000051200000000014507145975869
83486784401774840978007748409780004591650240000178564176000004761711360000088179840000000111974400000000933120000000004608000000000010240000000000420707233300201
9313810596097670925682208523250758000568216717200002525407632000007856823744000001745960832000000277136640000000030792960000000002280960000000000101376000000000020480000000000012200509765705829
102824295364817531454306160920511081864006818600606400003409300303200000121219566336000003142729497600000059861514240000000831409920000000008211456000000000054743040000000000221184000000000004096000000000000353814783205469041
11254186582832973431679485060979089059800800797776270948800044320903941600000177283615766400000525284787456000000116729952768000000019454992128000000002401850880000000000213497856000000000012939264000000000004792320000000000008192000000000000010260628712958602189
1222876792454961711722431932120102804351279084009138164558140800055844338966416000024819706207296000008273235402432000000210113914982400000004085548346880000000060526642176000000000672518246400000000005434490880000000000030191616000000000000103219200000000000001638400000000000000297558232675799463481
13205891132094649686303773648830010675836478981800010280435127908400006853623418605600000335066033798496000001240985310364800000003545672315328000000007879271811840000000001361849448960000000000181579926528000000000018341406720000000000001358622720000000000000696729600000000000000221184000000000000000327680000000000000008629188747598184440949
141853020188851841658851622702876801098086037838128000113875589109139200008224348102326720000043863189879075840000017870188469253120000005673075704524800000000141826892613120000000002801518866432000000000043579182366720000000000528232513536000000000004891041792000000000000033443020800000000000000159252480000000000000004718592000000000000000655360000000000000000250246473680347348787521
Technical details
Node id
n-0077d10b6afc996c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9eb007f6301418ee

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:          81,          360,           400;
  1:         729,         4860,         10800,           8000;
  2:        6561,        58320,        194400,         288000,          160000;
  3:       59049,       656100,       2916000,        6480000,         7200000,         3200000;
  4:      531441,      7085880,      39366000,      116640000,       194400000,       172800000,         64000000;
  5:     4782969,     74401740,     496011600,     1837080000,      4082400000,      5443200000,       4032000000,       1280000000;
  6:    43046721,    765275040,    5952139200,    26453952000,     73483200000,    130636800000,     145152000000,      92160000000,      25600000000;
  7:   387420489,   7748409780,   68874753600,   357128352000,   1190427840000,   2645395200000,    3919104000000,    3732480000000,    2073600000000,     512000000000;
  8:  3486784401,  77484097800,  774840978000,  4591650240000,  17856417600000,  47617113600000,   88179840000000,  111974400000000,   93312000000000,   46080000000000,   10240000000000;
  9: 31381059609, 767092568220, 8523250758000, 56821671720000, 252540763200000, 785682374400000, 1745960832000000, 2771366400000000, 3079296000000000, 2280960000000000, 1013760000000000, 204800000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (9+29*x+20*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 81 + 360*x + 400*x^2.
%t Mathematica
P[0, x_] := 81 + 360*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9 + 29*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] = 81 + 360*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((9 + 29*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.