Real-rooted atlas

A139548: Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.

A139548 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A139548
\(P_{n}(x) = \left(2 + 4\,x + 2\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12428
24162416464
3848120160120488512
4161284488961120896448128164096
53232014403840672080646720384014403203232768
664768422414080316805068859136506883168014080422476864262144
712817921164846592128128256256384384439296384384256256128128465921164817921282097152

Show all 15 stored rows

Generated b-files: A001018 (row sums), A139548, A139548 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-336459bdce627a87 · revision 1 · JSON
Definition source
oeis_catalog:A139548
Fingerprint
f6-2cd9f46ca2d53115

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: 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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2,    4,     2;
  2:   4,   16,    24,     16,       4;
  3:   8,   48,   120,    160,     120,      48,       8;
  4:  16,  128,   448,    896,    1120,     896,     448,      128,       16;
  5:  32,  320,  1440,   3840,    6720,    8064,    6720,     3840,     1440,      320,       32;
  6:  64,  768,  4224,  14080,   31680,   50688,   59136,    50688,    31680,    14080,     4224,      768,      64;
  7: 128, 1792, 11648,  46592,  128128,  256256,  384384,   439296,   384384,   256256,   128128,    46592,   11648,    1792,     128;
  8: 256, 4096, 30720, 143360,  465920, 1118208, 2050048,  2928640,  3294720,  2928640,  2050048,  1118208,  465920,  143360,   30720,   4096,   256;
  9: 512, 9216, 78336, 417792, 1566720, 4386816, 9504768, 16293888, 22404096, 24893440, 22404096, 16293888, 9504768, 4386816, 1566720, 417792, 78336, 9216, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+4*x+2*x^2)*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[(2 + 4*x + 2*x^2)*P[n - 1, 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] = (2 + 4*x + 2*x^2)*v[i - 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)

Generated with polytool c48629374597.

A139548: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,    4,     2;
  2:   4,   16,    24,     16,       4;
  3:   8,   48,   120,    160,     120,      48,       8;
  4:  16,  128,   448,    896,    1120,     896,     448,      128,       16;
  5:  32,  320,  1440,   3840,    6720,    8064,    6720,     3840,     1440,      320,       32;
  6:  64,  768,  4224,  14080,   31680,   50688,   59136,    50688,    31680,    14080,     4224,      768,      64;
  7: 128, 1792, 11648,  46592,  128128,  256256,  384384,   439296,   384384,   256256,   128128,    46592,   11648,    1792,     128;
  8: 256, 4096, 30720, 143360,  465920, 1118208, 2050048,  2928640,  3294720,  2928640,  2050048,  1118208,  465920,  143360,   30720,   4096,   256;
  9: 512, 9216, 78336, 417792, 1566720, 4386816, 9504768, 16293888, 22404096, 24893440, 22404096, 16293888, 9504768, 4386816, 1566720, 417792, 78336, 9216, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+4*x+2*x^2)*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[(2 + 4*x + 2*x^2)*P[n - 1, 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] = (2 + 4*x + 2*x^2)*v[i - 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)

Generated with polytool c48629374597.

A139548: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,    4,     2;
  2:   4,   16,    24,     16,       4;
  3:   8,   48,   120,    160,     120,      48,       8;
  4:  16,  128,   448,    896,    1120,     896,     448,      128,       16;
  5:  32,  320,  1440,   3840,    6720,    8064,    6720,     3840,     1440,      320,       32;
  6:  64,  768,  4224,  14080,   31680,   50688,   59136,    50688,    31680,    14080,     4224,      768,      64;
  7: 128, 1792, 11648,  46592,  128128,  256256,  384384,   439296,   384384,   256256,   128128,    46592,   11648,    1792,     128;
  8: 256, 4096, 30720, 143360,  465920, 1118208, 2050048,  2928640,  3294720,  2928640,  2050048,  1118208,  465920,  143360,   30720,   4096,   256;
  9: 512, 9216, 78336, 417792, 1566720, 4386816, 9504768, 16293888, 22404096, 24893440, 22404096, 16293888, 9504768, 4386816, 1566720, 417792, 78336, 9216, 512;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+4*x+2*x^2)*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[(2 + 4*x + 2*x^2)*P[n - 1, x]];
Table[Reverse[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] = (2 + 4*x + 2*x^2)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-4755dd2fd7289161
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-4755dd2fd7289161
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-4755dd2fd7289161

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 6 proof steps

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.