Real-rooted atlas

A013612 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(26 + 20\,x\right)\,P_{n-1}(x) + \left(20\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 676 + 1440\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
067614404002516
11757679760552008000160536
245697640204805238400179200016000011667856
311881376194081600426144000258880000528000003200000946986976
4308915776916538112032007136000307865600001077440000014592000006400000084565592896
580318101764277858470402295778598400328778688000017578176000003993472000003852800000012800000008216355935616
620882706457619838785167360160077104435200328665243648000252084403200000854740992000001361203200000098304000000025600000000860969134715136
754295036789769167606589900801096186459607040031466753042432000332942516121600001581142456320000036968386560000004354252800000002442240000000051200000000096613682312531456

All 15 rows

Technical details
Node id
n-4382590f97f7a0e0 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5b5479335e540f4e

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:              676,                1440,                  400;
  1:            17576,               79760,                55200,                  8000;
  2:           456976,             4020480,              5238400,               1792000,                160000;
  3:         11881376,           194081600,            426144000,             258880000,              52800000,               3200000;
  4:        308915776,          9165381120,          32007136000,           30786560000,           10774400000,            1459200000,              64000000;
  5:       8031810176,        427785847040,        2295778598400,         3287786880000,         1757817600000,          399347200000,           38528000000,           1280000000;
  6:     208827064576,      19838785167360,      160077104435200,       328665243648000,       252084403200000,        85474099200000,        13612032000000,         983040000000,         25600000000;
  7:    5429503678976,     916760658990080,    10961864596070400,     31466753042432000,     33294251612160000,     15811424563200000,      3696838656000000,      435425280000000,      24422400000000,       512000000000;
  8:  141167095653376,   42279580387123200,   741818276520448000,   2925378053570560000,   4158525731737600000,   2658124527206400000,    855966935040000000,   146217369600000000,   13251072000000000,    593920000000000,    10240000000000;
  9: 3670344486987776, 1947684039720734720, 49805597858092032000, 266418878137477120000, 499311288635596800000, 418094205062758400000, 178133663059968000000, 41391422054400000000, 5389046784000000000, 387368960000000000, 14192640000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = (26+20*x)*P(n-1,x) + 20*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 676 + 1440*x + 400*x^2.
%t Mathematica
P[0, x_] := 676 + 1440*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(26 + 20*x)*P[n - 1, x] + 20*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] = 676 + 1440*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 20*x)*v[i - 1] + 20*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

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.