Real-rooted atlas

A093644: (9,1) Pascal triangle.

A093644 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A093644

\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 - x + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19110
2910120
391911140
49283012180
59375842131160
69469510055141320
795514119515569151640

All 15 rows b-files: A093644 ?

Technical details
Node id
n-2bc3c94571dff60c · revision 1 · JSON
Definition source
oeis_catalog:A093644
Fingerprint
f6-e75dc09dd574571b

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): yes

Matches ?

  • A093644 triangle 69 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A020714, A257113, A146523, A287798, A084215, A323831

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 9,  1;
  2: 9, 10,   1;
  3: 9, 19,  11,   1;
  4: 9, 28,  30,  12,   1;
  5: 9, 37,  58,  42,  13,   1;
  6: 9, 46,  95, 100,  55,  14,   1;
  7: 9, 55, 141, 195, 155,  69,  15,   1;
  8: 9, 64, 196, 336, 350, 224,  84,  16,  1;
  9: 9, 73, 260, 532, 686, 574, 308, 100, 17, 1;
%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) = 9 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 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] = 9 + 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

A093644 triangle, offset 0, from P0
%e Example
Triangle begins:
  0: 1;
  1: 9,  1;
  2: 9, 10,   1;
  3: 9, 19,  11,   1;
  4: 9, 28,  30,  12,   1;
  5: 9, 37,  58,  42,  13,   1;
  6: 9, 46,  95, 100,  55,  14,   1;
  7: 9, 55, 141, 195, 155,  69,  15,   1;
  8: 9, 64, 196, 336, 350, 224,  84,  16,  1;
  9: 9, 73, 260, 532, 686, 574, 308, 100, 17, 1;
%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) = 9 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 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] = 9 + 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

Properties

Proof chains ? JSON

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.