Real-rooted atlas

A038336 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(22 + 54\,x + 32\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 1408\,x + 1024\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0484140810242916
110648464646758432768157464
223425613629442973696288358410485768503056
351536323748096010903552015859712011534336033554432459165024
41133799049894973443598172160697827328076126617604429185024107374182424794911296
5249435788825397098496110823702528268663521280390783303680341047246848165356240896343597383681338925209984
654875873536638555619328325082860748894569559490561719446536192020008105148416145513491988486047313952768109951162777672301961339136
712072692177921580425157836891952009183232312079546318848680900828332032990401204846592960389047123968598684081324032217703302299648351843720888323904305912313344

All 15 rows

Technical details
Node id
n-567e404cc344d278 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-89a932e93fe55d02

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:             484,             1408,              1024;
  1:           10648,            46464,             67584,              32768;
  2:          234256,          1362944,           2973696,            2883584,            1048576;
  3:         5153632,         37480960,         109035520,          158597120,          115343360,            33554432;
  4:       113379904,        989497344,        3598172160,         6978273280,         7612661760,          4429185024,          1073741824;
  5:      2494357888,      25397098496,      110823702528,       268663521280,       390783303680,        341047246848,        165356240896,         34359738368;
  6:     54875873536,     638555619328,     3250828607488,      9456955949056,     17194465361920,      20008105148416,      14551349198848,       6047313952768,       1099511627776;
  7:   1207269217792,   15804251578368,    91952009183232,    312079546318848,    680900828332032,     990401204846592,     960389047123968,     598684081324032,     217703302299648,     35184372088832;
  8:  26559922791424,  386326149693440,  2528680252538880,   9808214312878080,  24966363705507840,   43577653013250048,   52821397591818240,   43903499297095680,   23947363252961280,   7740561859543040,   1125899906842624;
  9: 584318301411328, 9349092822581248, 67993402346045440, 296698482964561920, 863122859533271040, 1757632004867751936, 2556555643444002816, 2656161707474288640, 1931753969072209920, 936607985004707840, 272467777455915008, 36028797018963968;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (22+54*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1408*x + 1024*x^2.
%t Mathematica
P[0, x_] := 484 + 1408*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((22 + 54*x + 32*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] = 484 + 1408*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((22 + 54*x + 32*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.