Real-rooted atlas

A225470 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
2191828
311899162280
4130315135019443640
51457656075221942916058240
61631575198451315444286525248801106560
7184289852920548289319107695265721102248024344320
81108491412247218183691635001286112396239660208264539520608608000
91135783025515051251136544597552756272025646949006735365136714256704017041024000
101165118804900501277961321919936524909419701839157650083676212136209203521120214277011200528271744000
1111981732588209028951263640926594972452101510059266151069059823663629705185216087117993208160707114136960017961239296000
12123424453150579060706503168317206232797878399450675418050431193405099627832055040504114056659986048263318896863360254561089305600664565853952000
1312733357924594571194323134050725679984415888171729792675611218882753549461959974830293481199506806565704471152863631923210523998066976640992788248291840026582634158080000
141315450453869775222729507906688282526857206733558643394059259453956773060811153050479370809431401093798320550908145120788002084082007923843844519358012959372804169710642825728001143053268797440000
Technical details
Node id
n-58dcdbbd8383c083 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-bc84b3931f09589a

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: 1,   3;
  2: 1,   9,   18;
  3: 1,  18,   99,    162;
  4: 1,  30,  315,   1350,    1944;
  5: 1,  45,  765,   6075,   22194,    29160;
  6: 1,  63, 1575,  19845,  131544,   428652,    524880;
  7: 1,  84, 2898,  52920,  548289,  3191076,   9526572,   11022480;
  8: 1, 108, 4914, 122472, 1818369, 16350012,  86112396,  239660208,  264539520;
  9: 1, 135, 7830, 255150, 5125113, 65445975, 527562720, 2564694900, 6735365136, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*n*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[(1 + 3*n*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] = (1 + 3*n*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.