Real-rooted atlas

A038321 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(18 + 43\,x + 25\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 900\,x + 625\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03249006251849
1583224300337501562579507
2104976583200121500011250003906253418801
31889568131220003645000050625000351562509765625147008443
4340122242834352009841500001822500000189843750010546875002441406256321363049
5612220032595213920024800580000574087500007973437500066445312500307617187506103515625271818611107
611019960576122444006400595213920000165337200000028704375000003189375000000221484375000087890625000015258789062511688200277601
7198359290368247949112960013774950720000446410440000009300217500000012916968750000011960156250000071191406250000247192382812503814697265625502592611936843

All 15 rows

Technical details
Node id
n-a3a321fd773cf3ba · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9fb853a77717bbd4

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:            324,             900,              625;
  1:           5832,           24300,            33750,             15625;
  2:         104976,          583200,          1215000,           1125000,            390625;
  3:        1889568,        13122000,         36450000,          50625000,          35156250,            9765625;
  4:       34012224,       283435200,        984150000,        1822500000,        1898437500,         1054687500,          244140625;
  5:      612220032,      5952139200,      24800580000,       57408750000,       79734375000,        66445312500,        30761718750,         6103515625;
  6:    11019960576,    122444006400,     595213920000,     1653372000000,     2870437500000,      3189375000000,      2214843750000,       878906250000,       152587890625;
  7:   198359290368,   2479491129600,   13774950720000,    44641044000000,    93002175000000,    129169687500000,    119601562500000,     71191406250000,     24719238281250,     3814697265625;
  8:  3570467226624,  49589822592000,  309936391200000,  1147912560000000,  2790065250000000,   4650108750000000,   5382070312500000,   4271484375000000,   2224731445312500,   686645507812500,    95367431640625;
  9: 64268410079232, 981878487321600, 6818600606400000, 28410835860000000, 78918988500000000, 153453588750000000, 213129984375000000, 211438476562500000, 146832275390625000, 67977905273437500, 18882751464843750, 2384185791015625;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (18+43*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 900*x + 625*x^2.
%t Mathematica
P[0, x_] := 324 + 900*x + 625*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18 + 43*x + 25*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] = 324 + 900*x + 625*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + 43*x + 25*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.