Real-rooted atlas

A038336 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(42 + 40\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1764 + 3360\,x + 1600\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01764336016006724
17408821168020160064000551368
23111696118540801693440010752000256000045212176
3130691232622339200118540800011289600005376000001024000003707398432
4548903174431365895680746807040009483264000067737600000258048000004096000000304006671424
523053933324815369288883204391225395200697019904000066382848000003793305600000120422400000016384000000024928547056768
69682651996416737725866393602459086221312004683973754880005576159232000004248502272000002023096320000005505024000000065536000000002044140858654976
740667138384947234857547187097601327906559508480029509034655744000421557637939200004014834647040000025491013632000000104044953600000002477260800000000262144000000000167619550409708032

All 15 rows

Technical details
Node id
n-10de7fc69fcc7b29 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-640a8dc8403e6c62

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,                3360,                 1600;
  1:              74088,              211680,               201600,                 64000;
  2:            3111696,            11854080,             16934400,              10752000,               2560000;
  3:          130691232,           622339200,           1185408000,            1128960000,             537600000,             102400000;
  4:         5489031744,         31365895680,          74680704000,           94832640000,           67737600000,           25804800000,            4096000000;
  5:       230539333248,       1536928888320,        4391225395200,         6970199040000,         6638284800000,         3793305600000,         1204224000000,          163840000000;
  6:      9682651996416,      73772586639360,      245908622131200,       468397375488000,       557615923200000,       424850227200000,       202309632000000,        55050240000000,        6553600000000;
  7:    406671383849472,    3485754718709760,    13279065595084800,     29509034655744000,     42155763793920000,     40148346470400000,     25491013632000000,     10404495360000000,     2477260800000000,      262144000000000;
  8:  17080198121677824,  162668553539788800,   697150943741952000,   1770542079344640000,   2950903465574400000,   3372461103513600000,   2676556431360000000,   1456629350400000000,   520224768000000000,   110100480000000000,   10485760000000000;
  9: 717368321110468608, 7515287173538242560, 35787081778753536000, 102248805082152960000, 194759628727910400000, 259679504970547200000, 247313814257664000000, 168240689971200000000, 80114614272000000000, 25433210880000000000, 4844421120000000000, 419430400000000000;
%F Formula
The row polynomials satisfy P(n,x) = (42+40*x)*P(n-1,x) for n >= 1, with P(0,x) = 1764 + 3360*x + 1600*x^2.
%t Mathematica
P[0, x_] := 1764 + 3360*x + 1600*x^2;
P[n_, x_] := P[n, x] = Expand[(42 + 40*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 + 3360*x + 1600*x^2; for(n = 1, N, my(i = n + 1); v[i] = (42 + 40*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.