Real-rooted atlas

A038336 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(42 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1764 + 1680\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0176416804003844
174088105840504008000238328
2311169659270404233600134400016000014776336
3130691232311169600296352000141120000336000003200000916132832
4548903174415682947840186701760001185408000042336000008064000006400000056800235584
523053933324876846444416010978063488008712748800004148928000001185408000001881600000012800000003521614606208
696826519964163688629331968061477155532800585496719360003485099520000013276569600000316108800000043008000000025600000000218340105584896
74066713838494721742877359354880331976639877120036886293319680002634735237120000125463582720000039829708800000081285120000000967680000000051200000000013537086546263552

All 15 rows

Technical details
Node id
n-3c5688d9dc1ec135 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-09bb967aa8554943

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:               1764,                1680,                 400;
  1:              74088,              105840,               50400,                 8000;
  2:            3111696,             5927040,             4233600,              1344000,               160000;
  3:          130691232,           311169600,           296352000,            141120000,             33600000,             3200000;
  4:         5489031744,         15682947840,         18670176000,          11854080000,           4233600000,           806400000,            64000000;
  5:       230539333248,        768464444160,       1097806348800,         871274880000,         414892800000,        118540800000,         18816000000,          1280000000;
  6:      9682651996416,      36886293319680,      61477155532800,       58549671936000,       34850995200000,      13276569600000,       3161088000000,        430080000000,        25600000000;
  7:    406671383849472,    1742877359354880,    3319766398771200,     3688629331968000,     2634735237120000,    1254635827200000,     398297088000000,      81285120000000,      9676800000000,      512000000000;
  8:  17080198121677824,   81334276769894400,  174287735935488000,   221317759918080000,   184431466598400000,  105389409484800000,   41821194240000000,   11379916800000000,   2032128000000000,   215040000000000,   10240000000000;
  9: 717368321110468608, 3757643586769121280, 8946770444688384000, 12781100635269120000, 12172476795494400000, 8114984530329600000, 3864278347776000000, 1314380390400000000, 312947712000000000, 49674240000000000, 4730880000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = (42+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1764 + 1680*x + 400*x^2.
%t Mathematica
P[0, x_] := 1764 + 1680*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(42 + 20*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] = 1764 + 1680*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (42 + 20*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.