Real-rooted atlas

veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22Pn(1)
1011
2000112
3000001214
4000000013318
50000000001464116
6000000000001510105132
70000000000000161520156164
8000000000000000172135352171128

All 15 rows b-files: A000079 (row sums), A007318, A007318 (reversed triangle) ?

Technical details
Node id
n-07e2f0a7469aac14 · revision 0 · JSON
Definition source
derived:veronese_parent_of:oeis_catalog:A098172
Fingerprint
f6-77a7ed3b1d5f257e

Observed on the stored rows ?

  • Degrees: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43 (affine: slope 3, intercept -2)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

  • A007318 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?
  • A007318 reversed triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A050732, A034008, A118655, A220051, A011782, A166444, A131577, A000079

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 0, 1;
   2: 0, 0, 0, 1, 1;
   3: 0, 0, 0, 0, 0, 1, 2, 1;
   4: 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1;
   5: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 6, 4,  1;
   6: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 10, 10,  5,  1;
   7: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  1,  6, 15, 20, 15,  6,  1;
   8: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  1,  7, 21, 35, 35, 21,  7,  1;
   9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  1,  8, 28, 56, 70, 56,  28,   8,  1;
  10: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  1,  9, 36, 84, 126, 126, 84, 36, 9, 1;
%F Formula
The row polynomials satisfy P(n,x) = (x^2+x^3)*P(n-1,x) for n >= 2, with P(1,x) = x.
%t Mathematica
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[(x^2 + x^3)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = (x^2 + x^3)*v[i - 1]); 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

A007318 triangle, offset 0, from P1, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1, 1;
  2: 1, 2,  1;
  3: 1, 3,  3,  1;
  4: 1, 4,  6,  4,   1;
  5: 1, 5, 10, 10,   5,   1;
  6: 1, 6, 15, 20,  15,   6,  1;
  7: 1, 7, 21, 35,  35,  21,  7,  1;
  8: 1, 8, 28, 56,  70,  56, 28,  8, 1;
  9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = x.
%t Mathematica
P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[(x^2 + x^3)*P[n - 1, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = (x^2 + x^3)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A007318 reversed triangle, offset 0, from P1, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1, 1;
  2: 1, 2,  1;
  3: 1, 3,  3,  1;
  4: 1, 4,  6,  4,   1;
  5: 1, 5, 10, 10,   5,   1;
  6: 1, 6, 15, 20,  15,   6,  1;
  7: 1, 7, 21, 35,  35,  21,  7,  1;
  8: 1, 8, 28, 56,  70,  56, 28,  8, 1;
  9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = (x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = x.
%t Mathematica
P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[(x^2 + x^3)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = (x^2 + x^3)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 1

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.