Real-rooted atlas

A075501 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 12\,x + 6\,n x\right)\,P_{n-1}(x) + \left(-6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 18\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01183655
1136252216505
2160900324012965497
319023401944040176777669823
41126504075600390096489888466561007407
51168957622680022044967511616592531227993616157905
6121616632571536900849660419520141134400713836801679616284214097
712702700012700802958379233067440015912028802611802880858283776100776965432922775

All 15 rows

Technical details
Node id
n-933e48ee8d422802 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-1bf2dd853fa9ec9a

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: 1,  18,    36;
  1: 1,  36,   252,     216;
  2: 1,  60,   900,    3240,      1296;
  3: 1,  90,  2340,   19440,     40176,       7776;
  4: 1, 126,  5040,   75600,    390096,     489888,       46656;
  5: 1, 168,  9576,  226800,   2204496,    7511616,     5925312,       279936;
  6: 1, 216, 16632,  571536,   9008496,   60419520,   141134400,     71383680,       1679616;
  7: 1, 270, 27000, 1270080,  29583792,  330674400,  1591202880,   2611802880,     858283776,     10077696;
  8: 1, 330, 41580, 2566080,  82927152, 1395690912, 11511434880,  40800672000,   47870735616,  10309483008,     60466176;
  9: 1, 396, 61380, 4811400, 206098992, 4878631296, 61756307712, 386143718400, 1027086863616, 871982724096, 123774262272, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (1+(6*n+12)*x)*P(n-1,x) - 6*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 18*x + 36*x^2.
%t Mathematica
P[0, x_] := 1 + 18*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (6*n + 12)*x)*P[n - 1, x] - 6*x^2*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] = 1 + 18*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (6*n + 12)*x)*v[i - 1] - 6*x^2*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

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.