Real-rooted atlas

A038338 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(48 + 24\,x\right)\,P_{n-1}(x) + \left(24\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2304 + 2880\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0230428805765760
111059226265612441613824511488
253084162156544018247680464486433177650098176
32548039681680113664226934784099532800015925248079626245366808576
412230590464127083479040258180120576173903708160468202291205159780352191102976623569010688
558706834227294435446620162783529507225627064767873024109158019891201990528598016160526499840458647142478002119507968
62817928042905669402485587968028988333988249603915819226497024222142991538585659638805220556878593774321664484331382374411007531417610438221892681728
713526054605946885064609235363430429494460282830848053947030875144192041386556925149184015350751073979596829403317922693120293240636964864014265760717209626418075402241486267713092321280

All 15 rows

Technical details
Node id
n-bc9618541a3b490f · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:                2304,                  2880,                    576;
  1:              110592,                262656,                 124416,                  13824;
  2:             5308416,              21565440,               18247680,                4644864,                  331776;
  3:           254803968,            1680113664,             2269347840,              995328000,               159252480,                7962624;
  4:         12230590464,          127083479040,           258180120576,           173903708160,             46820229120,             5159780352,              191102976;
  5:        587068342272,         9443544662016,         27835295072256,         27064767873024,          10915801989120,          1990528598016,           160526499840,            4586471424;
  6:      28179280429056,       694024855879680,       2898833398824960,       3915819226497024,        2221429915385856,        596388052205568,         78593774321664,         4843313823744,         110075314176;
  7:    1352605460594688,     50646092353634304,     294944602828308480,     539470308751441920,      413865569251491840,     153507510739795968,      29403317922693120,      2932406369648640,      142657607172096,       2641807540224;
  8:   64925062108545024,   3678981180515942400,   29530188088004837376,   71815107518052433920,    72543929382249431040,   35722035466321526784,    9329617298912182272,   1339079405988741120,   104615578592870400,    4121219762749440,     63403380965376;
  9: 3116402981210161152, 266444846487752933376, 2923193604780847005696, 9326537416278408167424, 12169888411477176483840, 7742356263516002844672, 2648615372582855639040, 513151966867460456448, 57245644606018682880, 3598775903594741760, 117169448024014848, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (48+24*x)*P(n-1,x) + 24*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2304 + 2880*x + 576*x^2.
%t Mathematica
P[0, x_] := 2304 + 2880*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(48 + 24*x)*P[n - 1, x] + 24*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2304 + 2880*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (48 + 24*x)*v[i - 1] + 24*x*deriv(v[i - 1], x)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.