Real-rooted atlas

A038334 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(12 + 32\,x + 20\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 480\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01444804001024
11728864014400800032768
2207361382403456003840001600001048576
324883220736006912000115200009600000320000033554432
4298598429859840124416000276480000345600000230400000640000001073741824
53583180841803776020901888005806080000967680000096768000005376000000128000000034359738368
6429981696573308928033443020800111476736000232243200000309657600000258048000000122880000000256000000001099511627776
751597803527739670528051597803520020065812480005016453120000836075520000092897280000006635520000000276480000000051200000000035184372088832
861917364224103195607040077396705280003439853568000010032906240000020065812480000027869184000000026542080000000016588800000000061440000000000102400000000001125899906842624
974300837068813621820129280113515167744000567575838720000189191946240000044144787456000007357464576000000875888640000000072990720000000004055040000000000135168000000000020480000000000036028797018963968
108916100448256178322008965120163461841551360090812134195200003405455032320000090812134195200000176579149824000000252255928320000000262766592000000000194641920000000000973209600000000002949120000000000040960000000000001152921504606846976
1110699320537907223181861165465602318186116546560014166692934451200059027887226880000017708366168064000003935192481792000000655865413632000000081983176704000000007591034880000000000506068992000000000023003136000000000006389760000000000008192000000000000036893488147419103232
12128391846454886429958097506140160324546056316518400216364037544345600099166850541158400003305561684705280000082639042117632000000157407699271680000000229552894771200000000255058771968000000000212548976640000000000128817561600000000000536739840000000000001376256000000000000016384000000000000001180591620717411303424
1315407021574586368385175539364659200449371462592102400032454605631651840000162273028158259200000595001103246950400000165278084235264000000035416732336128000000005902788722688000000000765176315904000000000076517631590400000000005796790272000000000000322043904000000000000012386304000000000000002949120000000000000003276800000000000000037778931862957161709568
141848842588950364164930246903867637760616280862983454720004793295600982425600002596368450532147200000103854738021285888000003173339217317068800000075555695650406400000000141666929344512000000000209876932362240000000000244856421089280000000000222596746444800000000000154581073920000000000000792723456000000000000002831155200000000000000062914560000000000000006553600000000000000001208925819614629174706176
Technical details
Node id
n-4906612bf3a39f4d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b1029f17d1b64336

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:          144,            480,             400;
  1:         1728,           8640,           14400,            8000;
  2:        20736,         138240,          345600,          384000,           160000;
  3:       248832,        2073600,         6912000,        11520000,          9600000,          3200000;
  4:      2985984,       29859840,       124416000,       276480000,        345600000,        230400000,         64000000;
  5:     35831808,      418037760,      2090188800,      5806080000,       9676800000,       9676800000,       5376000000,       1280000000;
  6:    429981696,     5733089280,     33443020800,    111476736000,     232243200000,     309657600000,     258048000000,     122880000000,      25600000000;
  7:   5159780352,    77396705280,    515978035200,   2006581248000,    5016453120000,    8360755200000,    9289728000000,    6635520000000,    2764800000000,     512000000000;
  8:  61917364224,  1031956070400,   7739670528000,  34398535680000,  100329062400000,  200658124800000,  278691840000000,  265420800000000,  165888000000000,   61440000000000,   10240000000000;
  9: 743008370688, 13621820129280, 113515167744000, 567575838720000, 1891919462400000, 4414478745600000, 7357464576000000, 8758886400000000, 7299072000000000, 4055040000000000, 1351680000000000, 204800000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (12+32*x+20*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 480*x + 400*x^2.
%t Mathematica
P[0, x_] := 144 + 480*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 32*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] = 144 + 480*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 32*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.