Real-rooted atlas

A038305 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(22 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 792\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04847923241600
1106482613621384583264000
22342567666569408965132161049762560000
35153632210830403449952028226880115473601889568102400000
411337990455659225611384841601241982720762125760249422976340122244096000000
5249435788814285867904350653121284781633472039122455680192055691525237882496612220032163840000000
6548758735363591875358721028582489088168313498214417213880499201126726723584460933659648107750725632110199605766553600000000
7120726921779288898915128322909419040563255543454410752681669667768325577297281740830421621536768106673218375682181952194048198359290368262144000000000

All 15 rows

Technical details
Node id
n-521827c5e15e2360 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-6bae7ffb6b8d3725

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:             484,              792,               324;
  1:           10648,            26136,             21384,              5832;
  2:          234256,           766656,            940896,            513216,            104976;
  3:         5153632,         21083040,          34499520,          28226880,          11547360,           1889568;
  4:       113379904,        556592256,        1138484160,        1241982720,         762125760,         249422976,          34012224;
  5:      2494357888,      14285867904,       35065312128,       47816334720,       39122455680,       19205569152,        5237882496,         612220032;
  6:     54875873536,     359187535872,     1028582489088,     1683134982144,     1721388049920,     1126726723584,      460933659648,      107750725632,       11019960576;
  7:   1207269217792,    8889891512832,    29094190405632,    55543454410752,    68166966776832,    55772972817408,    30421621536768,    10667321837568,     2181952194048,     198359290368;
  8:  26559922791424,  217308459202560,   800090236154880,  1745651424337920,  2499455448483840,  2454010803965952,  1673189184522240,   782270268088320,   240014741345280,   43639043880960,   3570467226624;
  9: 584318301411328, 5258864712701952, 21513537461053440, 52805955586222080, 86409745504727040, 98978435759960064, 80982356530876416, 47327351219343360, 19361189135185920, 5280324309596160, 864053068843008, 64268410079232;
%F Formula
The row polynomials satisfy P(n,x) = (22+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 792*x + 324*x^2.
%t Mathematica
P[0, x_] := 484 + 792*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 18*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] = 484 + 792*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 18*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.