Real-rooted atlas

A132200: Numbers in (4,4)-Pascal triangle .

A132200 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A132200

\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 - x + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
248416
341212432
44162416464
54204040204128
6424608060244256
74288414014084284512

All 15 rows b-files: A132200, A132200 (reversed triangle), A146541 (row sums) ?

Technical details
Node id
n-3aa21f78720c1d0c · revision 1 · JSON
Definition source
oeis_catalog:A132200
Fingerprint
f6-77a7ed3b1d5f257e

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: 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 ?

  • A132200 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
  • A132200 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A046055, A077552, A020707, A000079, A155559, A151821, A171449, A146541

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 4,  4;
  2: 4,  8,   4;
  3: 4, 12,  12,   4;
  4: 4, 16,  24,  16,   4;
  5: 4, 20,  40,  40,  20,   4;
  6: 4, 24,  60,  80,  60,  24,   4;
  7: 4, 28,  84, 140, 140,  84,  28,   4;
  8: 4, 32, 112, 224, 280, 224, 112,  32,  4;
  9: 4, 36, 144, 336, 504, 504, 336, 144, 36, 4;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (n - 1)*x)*v[i - 1])/(n - 1)); 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

A132200 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 4,  4;
  2: 4,  8,   4;
  3: 4, 12,  12,   4;
  4: 4, 16,  24,  16,   4;
  5: 4, 20,  40,  40,  20,   4;
  6: 4, 24,  60,  80,  60,  24,   4;
  7: 4, 28,  84, 140, 140,  84,  28,   4;
  8: 4, 32, 112, 224, 280, 224, 112,  32,  4;
  9: 4, 36, 144, 336, 504, 504, 336, 144, 36, 4;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (n - 1)*x)*v[i - 1])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); 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

A132200 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 4,  4;
  2: 4,  8,   4;
  3: 4, 12,  12,   4;
  4: 4, 16,  24,  16,   4;
  5: 4, 20,  40,  40,  20,   4;
  6: 4, 24,  60,  80,  60,  24,   4;
  7: 4, 28,  84, 140, 140,  84,  28,   4;
  8: 4, 32, 112, 224, 280, 224, 112,  32,  4;
  9: 4, 36, 144, 336, 504, 504, 336, 144, 36, 4;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (n - 1)*x)*v[i - 1])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); 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

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.