Real-rooted atlas

A143497 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(5 + x\right)\,P_{n-1}(x) + \left(6 + 2\,x\right)\,P_{n-1}'(x) + \left(1 + x\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 31 + 12\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03112144
1229129211380
2196114323423213768
31908116965529073045142112
420777521627682635149601365601523072
52501801296835713354112975353573523317717141248
63298996943761264225567165941376880950758243724961106209920
7472630861690761097399936132121142532212966462268126147252565211711708188416

All 15 rows

Technical details
Node id
n-b323b1948005800d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e10cc3609cf27a7c

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:           31,           12,            1;
  1:          229,          129,           21,           1;
  2:         1961,         1432,          342,          32,           1;
  3:        19081,        16965,         5290,         730,          45,          1;
  4:       207775,       216276,        82635,       14960,        1365,         60,         1;
  5:      2501801,      2968357,      1335411,      297535,       35735,       2331,        77,        1;
  6:     32989969,     43761264,     22556716,     5941376,      880950,      75824,      3724,       96,      1;
  7:    472630861,    690761097,    399936132,   121142532,    21296646,    2268126,    147252,     5652,    117,     1;
  8:   7307593151,  11633919580,   7453166805,  2544545760,   515822790,   65274552,   5252730,   266880,   8235,   140,   1;
  9: 121247816845, 208356640041, 145971392545, 55318281645, 12671551530, 1852830210, 177450042, 11189970, 457545, 11605, 165, 1;
%F Formula
The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + (6+2*x)*d/dx P(n-1,x) + (1+x)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 31 + 12*x + x^2.
%t Mathematica
P[0, x_] := 31 + 12*x + x^2;
P[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + (6 + 2*x)*D[P[n - 1, x], x] + (1 + x)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 31 + 12*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1] + (6 + 2*x)*deriv(v[i - 1], x) + (1 + x)*deriv(deriv(v[i - 1], x), 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

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.