Real-rooted atlas

A193728: Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+1)^n.

A193728 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A193728

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4 - 3\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
2810321
33264429147
4128352360162271029
5512179224961728594817203
62048870415360144007560210624350421
78192409608755210368073440311047290729352947

All 15 rows b-files: A169634 (row sums), A193728, A193729 (reversed triangle) ?

Technical details
Node id
n-e5e648370a9e4459 · revision 2 · JSON
Definition source
oeis_catalog:A193728
Fingerprint
f6-63d3891a7d0f6d93

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 ?

  • A193728 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
  • A193729 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A169634

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      2,      1;
  2:      8,     10,       3;
  3:     32,     64,      42,       9;
  4:    128,    352,     360,     162,      27;
  5:    512,   1792,    2496,    1728,     594,      81;
  6:   2048,   8704,   15360,   14400,    7560,    2106,     243;
  7:   8192,  40960,   87552,  103680,   73440,   31104,    7290,    729;
  8:  32768, 188416,  473088,  677376,  604800,  344736,  122472,  24786,  2187;
  9: 131072, 851968, 2457600, 4128768, 4451328, 3193344, 1524096, 466560, 83106, 6561;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (3*n - 3)*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (3*n - 3)*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

A193728 triangle, offset 0, from P0
%e Example
Triangle begins:
  0:      1;
  1:      2,      1;
  2:      8,     10,       3;
  3:     32,     64,      42,       9;
  4:    128,    352,     360,     162,      27;
  5:    512,   1792,    2496,    1728,     594,      81;
  6:   2048,   8704,   15360,   14400,    7560,    2106,     243;
  7:   8192,  40960,   87552,  103680,   73440,   31104,    7290,    729;
  8:  32768, 188416,  473088,  677376,  604800,  344736,  122472,  24786,  2187;
  9: 131072, 851968, 2457600, 4128768, 4451328, 3193344, 1524096, 466560, 83106, 6561;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (3*n - 3)*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (3*n - 3)*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

A193729 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0:    1;
  1:    1,     2;
  2:    3,    10,      8;
  3:    9,    42,     64,      32;
  4:   27,   162,    360,     352,     128;
  5:   81,   594,   1728,    2496,    1792,     512;
  6:  243,  2106,   7560,   14400,   15360,    8704,    2048;
  7:  729,  7290,  31104,   73440,  103680,   87552,   40960,    8192;
  8: 2187, 24786, 122472,  344736,  604800,  677376,  473088,  188416,  32768;
  9: 6561, 83106, 466560, 1524096, 3193344, 4451328, 4128768, 2457600, 851968, 131072;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (4*n-4+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (3*n - 3)*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (3*n - 3)*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

Used in 5

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.