Real-rooted atlas

A257609: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.

A257609 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A257609
\(P_{n}(x) = \left(2 + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2416424
3888888192
4164161056416161920
53218249664966418243223040
66476807622415462476224768064322560
71283161654950419992321999232549504316161285160960

Show all 15 stored rows

Generated b-files: A257609, A257609 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-41809911e05bb4ba · revision 1 · JSON
Definition source
oeis_catalog:A257609
Fingerprint
f6-605cb6decd8257f5

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): no

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,      2;
  2:   4,     16,        4;
  3:   8,     88,       88,         8;
  4:  16,    416,     1056,       416,        16;
  5:  32,   1824,     9664,      9664,      1824,        32;
  6:  64,   7680,    76224,    154624,     76224,      7680,        64;
  7: 128,  31616,   549504,   1999232,   1999232,    549504,     31616,      128;
  8: 256, 128512,  3739648,  22587904,  39984640,  22587904,   3739648,   128512,    256;
  9: 512, 518656, 24494080, 233058304, 670901248, 670901248, 233058304, 24494080, 518656, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*n*x)*P(n-1,x) + (2*x-2*x^2)*d/dx 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 + 2*n*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], 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 + 2*n*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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.

A257609: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,      2;
  2:   4,     16,        4;
  3:   8,     88,       88,         8;
  4:  16,    416,     1056,       416,        16;
  5:  32,   1824,     9664,      9664,      1824,        32;
  6:  64,   7680,    76224,    154624,     76224,      7680,        64;
  7: 128,  31616,   549504,   1999232,   1999232,    549504,     31616,      128;
  8: 256, 128512,  3739648,  22587904,  39984640,  22587904,   3739648,   128512,    256;
  9: 512, 518656, 24494080, 233058304, 670901248, 670901248, 233058304, 24494080, 518656, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*n*x)*P(n-1,x) + (2*x-2*x^2)*d/dx 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 + 2*n*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
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; for(n = 1, N, my(i = n + 1); v[i] = (2 + 2*n*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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)

Generated with polytool c48629374597.

A257609: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,      2;
  2:   4,     16,        4;
  3:   8,     88,       88,         8;
  4:  16,    416,     1056,       416,        16;
  5:  32,   1824,     9664,      9664,      1824,        32;
  6:  64,   7680,    76224,    154624,     76224,      7680,        64;
  7: 128,  31616,   549504,   1999232,   1999232,    549504,     31616,      128;
  8: 256, 128512,  3739648,  22587904,  39984640,  22587904,   3739648,   128512,    256;
  9: 512, 518656, 24494080, 233058304, 670901248, 670901248, 233058304, 24494080, 518656, 512;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*n*x)*P(n-1,x) + (2*x-2*x^2)*d/dx 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 + 2*n*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
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; for(n = 1, N, my(i = n + 1); v[i] = (2 + 2*n*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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)

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-7ce0c29875c331b6
  • 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-7ce0c29875c331b6, e-f2db3ed118a27f94
  • 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-7ce0c29875c331b6, e-f2db3ed118a27f94
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-7ce0c29875c331b6
  • 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-f2db3ed118a27f94

Proof chains

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

Used in 4 proof steps

Same fingerprint (6)

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

References

  1. [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