Real-rooted atlas

A013612 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(16 + 10\,x\right)\,P_{n-1}(x) + \left(10\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 256 + 420\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0256420100776
14096134807800100026376
265536391440415600124000100001006576
3104857610832800188760009860000180000010000042517376
4167772162921385607878640006423200001994000002460000010000001964099776
5268435456776337472031284489600374253600001758960000036176000003220000001000000098280859776
642949672962045320972801203875372800203441145600013592712000004146576000006064800000040800000001000000005285870693376
76871947673653607842022404538483439360010562168070400096463301760000409601136000008755824000000957360000000504000000001000000000303624018136576

All 15 rows

Technical details
Node id
n-f8e5ffc520f5b05b · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-1cfe5e80b35ba55f

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:            256,              420,               100;
  1:           4096,            13480,              7800,               1000;
  2:          65536,           391440,            415600,             124000,              10000;
  3:        1048576,         10832800,          18876000,            9860000,            1800000,             100000;
  4:       16777216,        292138560,         787864000,          642320000,          199400000,           24600000,            1000000;
  5:      268435456,       7763374720,       31284489600,        37425360000,        17589600000,         3617600000,          322000000,          10000000;
  6:     4294967296,     204532097280,     1203875372800,      2034411456000,      1359271200000,       414657600000,        60648000000,        4080000000,        100000000;
  7:    68719476736,    5360784202240,    45384834393600,    105621680704000,     96463301760000,     40960113600000,      8755824000000,      957360000000,      50400000000,      1000000000;
  8:  1099511627776,  140067584025600,  1687461880192000,   5312445656320000,   6458161705600000,   3668000515200000,   1075043760000000,   169891200000000,   14412000000000,    610000000000,   10000000000;
  9: 17592186044416, 3652752300943360, 62149303527168000, 261247118992640000, 414781512076800000, 306669651059200000, 118383330912000000, 25361080800000000, 3082464000000000, 208780000000000, 7260000000000, 100000000000;
%F Formula
The row polynomials satisfy P(n,x) = (16+10*x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 256 + 420*x + 100*x^2.
%t Mathematica
P[0, x_] := 256 + 420*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 10*x)*P[n - 1, x] + 10*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] = 256 + 420*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 10*x)*v[i - 1] + 10*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.