Real-rooted atlas

A038330 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(28 + 12\,x\right)\,P_{n-1}(x) + \left(12\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 784 + 816\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
07848161441744
1219524204817280172883008
261465619453441403136317952207364301824
3172103688518963296307200371865605391360248832241533952
448189030436141096966030249984353562624085598208086593536298598414607437824
51349292851215034707148835694231552029864307916810748215296017892016128133772083235831808946173116416
6377801998336617579800166420365165264896233964648529921175236057497628642832547843484762767362006581248042998169665300846018560
71057845595340825156554404659211330981697945601741755733770240117393698193408039308525332070469219026731008642908631859229410748006451597803524779967519129600

All 15 rows

Technical details
Node id
n-227fbd5a5a45daab · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-72277c8e823c26dc

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:              784,                816,                 144;
  1:            21952,              42048,               17280,                1728;
  2:           614656,            1945344,             1403136,              317952,               20736;
  3:         17210368,           85189632,            96307200,            37186560,             5391360,              248832;
  4:        481890304,         3614109696,          6030249984,          3535626240,           855982080,            86593536,             2985984;
  5:      13492928512,       150347071488,        356942315520,        298643079168,        107482152960,         17892016128,          1337720832,           35831808;
  6:     377801998336,      6175798001664,      20365165264896,      23396464852992,      11752360574976,       2864283254784,        348476276736,        20065812480,         429981696;
  7:   10578455953408,    251565544046592,    1133098169794560,    1741755733770240,    1173936981934080,     393085253320704,      69219026731008,      6429086318592,      294107480064,       5159780352;
  8:  296196766695424,  10189563233304576,   61939891357876224,  125069544998830080,  110120279432232960,   48678746075430912,   11638925712949248,   1550685988454400,   113618363351040,    4231019888640,    61917364224;
  9: 8293509467471872, 411136890532528128, 3343149109409218560, 8747729576219639808, 9869975776835665920, 5605173007824715776, 1748037524200095744, 313343939262283776, 32696908916981760, 1938839065067520, 59936008568832, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (28+12*x)*P(n-1,x) + 12*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 784 + 816*x + 144*x^2.
%t Mathematica
P[0, x_] := 784 + 816*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 12*x)*P[n - 1, x] + 12*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] = 784 + 816*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 12*x)*v[i - 1] + 12*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.