Real-rooted atlas

A178820 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15 + 18\,x + 5\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1202444
22506003601210
32500900010800432026620
42187510500018900015120045360512435
517500010500002520000302400018144004354569018856
613125009450000283500004536000040824000195955203919104148811124
7937500078750000283500000567000000680400000489888000195955200335923202338460520

All 15 rows

Technical details
Node id
n-e5535bb2038f7b5a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

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:        20,         24;
  2:       250,        600,         360;
  3:      2500,       9000,       10800,        4320;
  4:     21875,     105000,      189000,      151200,        45360;
  5:    175000,    1050000,     2520000,     3024000,      1814400,       435456;
  6:   1312500,    9450000,    28350000,    45360000,     40824000,     19595520,      3919104;
  7:   9375000,   78750000,   283500000,   567000000,    680400000,    489888000,    195955200,    33592320;
  8:  64453125,  618750000,  2598750000,  6237000000,   9355500000,   8981280000,   5388768000,  1847577600,   277136640;
  9: 429687500, 4640625000, 22275000000, 62370000000, 112266000000, 134719200000, 107775360000, 55427328000, 16628198400, 2217093120;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+15+(6*n+18)*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[Together[((5*n + 15 + (6*n + 18)*x)*P[n - 1, x])/(n)]];
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] = ((5*n + 15 + (6*n + 18)*x)*v[i - 1])/(n)); 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.