Real-rooted atlas

A038325 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 2\,x + 6\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(88 + 132\,x + 66\,x^{2} + 11\,x^{3} + 132\,n x + 110\,n x^{2} + 22\,n x^{3} + 22\,n^{2} x^{2} + 11\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224466
2484193610893509
310648638887187421296167706
42342561874048316245618740483660257510833
5515363251536320115956720103072640402627505797836321779898
611337990413605588483826571760453519616026573415007653143528680648913345169013
724943578883492101043211785841020817460505216013641019700058929205104133681993061247178944539833611042

All 15 rows

Technical details
Node id
n-47d6e3010dd27a9f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:             1;
  1:            22,             44;
  2:           484,           1936,           1089;
  3:         10648,          63888,          71874,           21296;
  4:        234256,        1874048,        3162456,         1874048,          366025;
  5:       5153632,       51536320,      115956720,       103072640,        40262750,         5797836;
  6:     113379904,     1360558848,     3826571760,      4535196160,      2657341500,       765314352,       86806489;
  7:    2494357888,    34921010432,   117858410208,    174605052160,    136410197000,     58929205104,    13368199306,     1247178944;
  8:   54875873536,   878013976576,  3457180032768,   6146097836032,   6002048668000,   3457180032768,  1176401538928,   219503494144,   17363069361;
  9: 1207269217792, 21730845920256, 97788806641152, 202821228589056, 237681127252800, 171130411622016, 77642501569248, 21730845920256, 3437887733478, 235794769100;
%F Formula
The row polynomials satisfy (4+(6*n-2)*x+n^2*x^2)*P(n,x) = (88+(132*n+132)*x+(22*n^2+110*n+66)*x^2+(11*n^2+22*n+11)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((88 + (132*n + 132)*x + (22*n^2 + 110*n + 66)*x^2 + (11*n^2 + 22*n + 11)*x^3)*P[n - 1, x])/(4 + (6*n - 2)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((88 + (132*n + 132)*x + (22*n^2 + 110*n + 66)*x^2 + (11*n^2 + 22*n + 11)*x^3)*v[i - 1])/(4 + (6*n - 2)*x + n^2*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.