Real-rooted atlas

A094262: Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

A094262 ↗ nonnegative coefficients

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

Recurrence from A094262

\(P_{n}(x) = \left(1 + 2\,x + x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11214
2161210332
3114611241317015416
41302408901830222616006301057552
5162841506016990352164706240796222256930945176128
611262772254101279534014368369761196532118219579571834902090090103955018624
712548821118524851361378978611315829236985363564309938808946303933351671670061365151351350135135168968192

All 15 rows b-files: A005172 (row sums), A094262 ?

Technical details
Node id
n-f712c326ca8e5415 · revision 0 · JSON
Definition source
oeis_catalog:A094262
Fingerprint
f6-5870953131fc996b

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

  • A094262 triangle 49 terms, 7 rows · tabf · nonn · catalog b-file?

Row sums ? A005172, A225170

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,    2,     1;
  2: 1,    6,    12,      10,        3;
  3: 1,   14,    61,     124,      131,        70,         15;
  4: 1,   30,   240,     890,     1830,      2226,       1600,        630,         105;
  5: 1,   62,   841,    5060,    16990,     35216,      47062,      40796,       22225,        6930,         945;
  6: 1,  126,  2772,   25410,   127953,    401436,     836976,    1196532,     1182195,      795718,      349020,       90090,       10395;
  7: 1,  254,  8821,  118524,   851361,   3789786,   11315829,   23698536,    35643099,    38808946,    30393335,    16716700,     6136515,     1351350,      135135;
  8: 1,  510, 27480,  527530,  5231460,  31726422,  128945040,  370748610,   779175270,  1219253530,  1431293496,  1257343230,   815302180,   379068690,   119639520,   22972950,    2027025;
  9: 1, 1022, 84481, 2276020, 30459980, 244783252, 1309489644, 4961577972, 13847170470, 29183679040, 47141876486, 58769110972, 56519394316, 41592937100, 23007444460, 9263714460, 2564186625, 436486050, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx 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[(1 + 2*x + x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*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; for(n = 1, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 1] + (x + 2*x^2 + x^3)*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

A094262 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1,    2,     1;
   3: 1,    6,    12,      10,        3;
   4: 1,   14,    61,     124,      131,        70,         15;
   5: 1,   30,   240,     890,     1830,      2226,       1600,        630,         105;
   6: 1,   62,   841,    5060,    16990,     35216,      47062,      40796,       22225,        6930,         945;
   7: 1,  126,  2772,   25410,   127953,    401436,     836976,    1196532,     1182195,      795718,      349020,       90090,       10395;
   8: 1,  254,  8821,  118524,   851361,   3789786,   11315829,   23698536,    35643099,    38808946,    30393335,    16716700,     6136515,     1351350,      135135;
   9: 1,  510, 27480,  527530,  5231460,  31726422,  128945040,  370748610,   779175270,  1219253530,  1431293496,  1257343230,   815302180,   379068690,   119639520,   22972950,    2027025;
  10: 1, 1022, 84481, 2276020, 30459980, 244783252, 1309489644, 4961577972, 13847170470, 29183679040, 47141876486, 58769110972, 56519394316, 41592937100, 23007444460, 9263714460, 2564186625, 436486050, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + 2*x + x^2)*v[i - 1] + (x + 2*x^2 + x^3)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 7

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.