Real-rooted atlas

A132393 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(4 + 4\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
230441690
321042828864990
4168042644016166425611880
51512045096532003104089601024154440
61512005114407123845232002137604608040962162160
71663200623064098819848604736444416013619202293761638432432400
8199584008142048014350636814278476887748864341196808200192111411265536518918400
925945920011382998402191264704243022745617118743047945512962430812164728422453084162621448821612800
103632428800169740345603523090521642788243200336871500801797121536065813422081634304000263454720249036801048576158789030400
11544864320002691402336005963597164807827472688646764602240004043168307201706049945605083992883210489036800142737408011534336041943043016991577600
1287178291200045241894656001061831639808014909395167744139543526594569174910187520434694723584014958588395523711843041286479413248075549900805284823041677721660339831552000
13148203095040008039835256320019860813662976029593298344396829686157588172821179088382566411059774375936042817389215744122935685283842586237468672387611361280392041594882399141888671088641267136462592000
142667655710720001506451584153600389653986958848061212262485104646527240299646976499968221238886428379229229711361213103980920832392553790373888957265485496321732195437772822561203159042000011919361080452710426843545627877002177024000
Technical details
Node id
n-3814a7624bdf369a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9005cb6408d407ba

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:         5,          4;
  2:        30,         44,         16;
  3:       210,        428,        288,         64;
  4:      1680,       4264,       4016,       1664,        256;
  5:     15120,      45096,      53200,      31040,       8960,      1024;
  6:    151200,     511440,     712384,     523200,     213760,     46080,      4096;
  7:   1663200,    6230640,    9881984,    8604736,    4444160,   1361920,    229376,    16384;
  8:  19958400,   81420480,  143506368,  142784768,   87748864,  34119680,   8200192,  1114112,   65536;
  9: 259459200, 1138299840, 2191264704, 2430227456, 1711874304, 794551296, 243081216, 47284224, 5308416, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (n+4+4*x)*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[(n + 4 + 4*x)*P[n - 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] = (n + 4 + 4*x)*v[i - 1]); 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.