Real-rooted atlas

A156992 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141014
224120150294
31921440360030008232
41920192007200012000075000288120
523040288000144000036000004500000225000012101040
632256048384003024000010080000018900000018900000078750000592950960
751609609031680067737600028224000007056000000105840000008820000000315000000033205253760
892897280185794560016257024000812851200002540160000005080320000006350400000004536000000001417500000002091930986880
918579456004180377600041803776000024385536000009144576000000228614400000003810240000000040824000000000255150000000007087500000000146435169081600
1040874803200102187008000011496038400000766402560000003353011200000001005903360000000209563200000000029937600000000002806650000000000155925000000000038981250000000011275508019283200
11980995276800269773701120003372171264000002529128448000000126456422400000004425974784000000011064936960000000019758816000000000024698520000000000020582100000000000010291050000000000023388750000000000947142673619788800
1225505877196800765176315904000105211743436800008767645286400000049318004736000000019727201894400000005753767219200000000123295011840000000001926484560000000000021405384000000000000160540380000000000007297290000000000000152026875000000000086189983299400780800
13714164561510400232103482490880003481552237363200003191422884249600000199463930265600000008975876861952000000029919589539840000000074798973849600000000014024807596800000000001947889944000000000000194788994400000000000013281067800000000000005533778250000000000001064188125000000000008446618363341276518400
1421424936845312000749872789585920000121854328307712000001218543283077120000008377485071155200000004188742535577600000000157077845084160000000004487938430976000000000098173653177600000000000163622755296000000000000204528444120000000000000185934949200000000000000116209343250000000000000446959012500000000000007981410937500000000000886894928150834034432000
Technical details
Node id
n-087054bdf6cd6b1d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9629d1e9c04dbfbe

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:          4,          10;
  2:         24,         120,          150;
  3:        192,        1440,         3600,          3000;
  4:       1920,       19200,        72000,        120000,         75000;
  5:      23040,      288000,      1440000,       3600000,       4500000,        2250000;
  6:     322560,     4838400,     30240000,     100800000,     189000000,      189000000,       78750000;
  7:    5160960,    90316800,    677376000,    2822400000,    7056000000,    10584000000,     8820000000,     3150000000;
  8:   92897280,  1857945600,  16257024000,   81285120000,  254016000000,   508032000000,   635040000000,   453600000000,   141750000000;
  9: 1857945600, 41803776000, 418037760000, 2438553600000, 9144576000000, 22861440000000, 38102400000000, 40824000000000, 25515000000000, 7087500000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+2+(7*n+7)*x+(5*n+5)*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 + 2 + (7*n + 7)*x + (5*n + 5)*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 + 2 + (7*n + 7)*x + (5*n + 5)*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.