Real-rooted atlas

A139548 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(50 + 60\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1506018128
2250060005400216032416384
31250004500006750005400002430005832058322097152
46250000300000006300000075600000567000002721600081648001399680104976268435456
5312500000187500000050625000008100000000850500000061236000003061800000104976000023619600031492800188956834359738368
6156250000001125000000003712500000007425000000001002375000000962280000000673596000000346420800000129907800000346420800006235574400680244480340122244398046511104
7781250000000656250000000025593750000000614250000000001013512500000001216215000000001094593500000007505784000000039405366000000157621464000004728643920000103170412800015475561920014285134080612220032562949953421312

All 15 rows

Technical details
Node id
n-211401dac7a27919 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-eb1af6eb566f4016

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:               50,                60,                 18;
  2:             2500,              6000,               5400,               2160,                324;
  3:           125000,            450000,             675000,             540000,             243000,               58320,                5832;
  4:          6250000,          30000000,           63000000,           75600000,           56700000,            27216000,             8164800,             1399680,              104976;
  5:        312500000,        1875000000,         5062500000,         8100000000,         8505000000,          6123600000,          3061800000,          1049760000,           236196000,           31492800,            1889568;
  6:      15625000000,      112500000000,       371250000000,       742500000000,      1002375000000,        962280000000,        673596000000,        346420800000,        129907800000,        34642080000,         6235574400,          680244480,          34012224;
  7:     781250000000,     6562500000000,     25593750000000,     61425000000000,    101351250000000,     121621500000000,     109459350000000,      75057840000000,      39405366000000,     15762146400000,      4728643920000,      1031704128000,      154755619200,       14285134080,        612220032;
  8:   39062500000000,   375000000000000,   1687500000000000,   4725000000000000,   9213750000000000,   13267800000000000,   14594580000000000,   12509640000000000,    8444007000000000,   4503470400000000,   1891457568000000,    619022476800000,   154755619200000,    28570268160000,    3673320192000,    293865615360,    11019960576;
  9: 1953125000000000, 21093750000000000, 107578125000000000, 344250000000000000, 774562500000000000, 1301265000000000000, 1691644500000000000, 1739977200000000000, 1435481190000000000, 956987460000000000, 516773228400000000, 225501045120000000, 78925365792000000, 21856255142400000, 4683483244800000, 749357319168000, 84302698406400, 5950778711040, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (50+60*x+18*x^2)*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[(50 + 60*x + 18*x^2)*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] = (50 + 60*x + 18*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.