Real-rooted atlas

A130534 → basis A008277 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(n + n x + n^{2}\right)\,P_{n-1}(x) + \left(n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2628
33624363
4240240604544
51800240090012055225
61512025200126002520210655656
71411202822401764004704058803367653023
81451520338688025401608467201411201209650488379008
9163296004354560038102400152409603175200362880226807209116780049
1019958400059875200059875200027941760069854400997920083160039600990101757211400
11263450880087816960009879408000526901760015367968002634508802744280017424006534013201128394129951
1237362124800136994457600171243072000102745843200342486144006849722880856215360679536003397680102960171612490371506208
1356665889280022666355712003116623910400207774927360077915597760017809279488025971865920247351104015459444062462401561562184139013522796473
14915372057600039666122496000594991837440004363273474560018180306144000467493586560077915597760086572886400649296648032792760010930920229320273014175679564492264
Technical details
Node id
n-c33cf77401a117b2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-caf342f7caa2ed5f

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        1;
  2:        6,        2;
  3:       36,       24,        3;
  4:      240,      240,       60,        4;
  5:     1800,     2400,      900,      120,       5;
  6:    15120,    25200,    12600,     2520,     210,      6;
  7:   141120,   282240,   176400,    47040,    5880,    336,     7;
  8:  1451520,  3386880,  2540160,   846720,  141120,  12096,   504,   8;
  9: 16329600, 43545600, 38102400, 15240960, 3175200, 362880, 22680, 720, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n^2+n+n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + n + n*x)*v[i - 1] + n*x*deriv(v[i - 1], x))/(n - 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. [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.