Real-rooted atlas

A257626: 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) = 3*x + 6.

A257626 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 A257626
\(P_{n}(x) = \left(6 + 3\,x + 3\,n x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16612
23610836180
3216140414042163240
41296158763369615876129668040
5777616621264249264249216621277761632960
646656165920410701720192747601070172016592044665644089920
727993616052580163263924481752360481752360163263924160525802799361322697600

Show all 15 stored rows

Generated b-files: A051609 (row sums), A257626, A257626 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-61cf3a32da4a0e22 · revision 1 · JSON
Definition source
oeis_catalog:A257626
Fingerprint
f6-9f31c05ba8767dc5

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:        6,          6;
  2:       36,        108,          36;
  3:      216,       1404,        1404,          216;
  4:     1296,      15876,       33696,        15876,         1296;
  5:     7776,     166212,      642492,       642492,       166212,         7776;
  6:    46656,    1659204,    10701720,     19274760,     10701720,      1659204,        46656;
  7:   279936,   16052580,   163263924,    481752360,    481752360,    163263924,     16052580,      279936;
  8:  1679616,  152031492,  2344429008,  10654827804,  17343084960,  10654827804,   2344429008,   152031492,    1679616;
  9: 10077696, 1418671908, 32237998380, 216088713252, 535926913164, 535926913164, 216088713252, 32237998380, 1418671908, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (6+(3*n+3)*x)*P(n-1,x) + (3*x-3*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[(6 + (3*n + 3)*x)*P[n - 1, x] + (3*x - 3*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] = (6 + (3*n + 3)*x)*v[i - 1] + (3*x - 3*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.

A257626: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,          6;
  2:       36,        108,          36;
  3:      216,       1404,        1404,          216;
  4:     1296,      15876,       33696,        15876,         1296;
  5:     7776,     166212,      642492,       642492,       166212,         7776;
  6:    46656,    1659204,    10701720,     19274760,     10701720,      1659204,        46656;
  7:   279936,   16052580,   163263924,    481752360,    481752360,    163263924,     16052580,      279936;
  8:  1679616,  152031492,  2344429008,  10654827804,  17343084960,  10654827804,   2344429008,   152031492,    1679616;
  9: 10077696, 1418671908, 32237998380, 216088713252, 535926913164, 535926913164, 216088713252, 32237998380, 1418671908, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (6+(3*n+3)*x)*P(n-1,x) + (3*x-3*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[(6 + (3*n + 3)*x)*P[n - 1, x] + (3*x - 3*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] = (6 + (3*n + 3)*x)*v[i - 1] + (3*x - 3*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.

A257626: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,          6;
  2:       36,        108,          36;
  3:      216,       1404,        1404,          216;
  4:     1296,      15876,       33696,        15876,         1296;
  5:     7776,     166212,      642492,       642492,       166212,         7776;
  6:    46656,    1659204,    10701720,     19274760,     10701720,      1659204,        46656;
  7:   279936,   16052580,   163263924,    481752360,    481752360,    163263924,     16052580,      279936;
  8:  1679616,  152031492,  2344429008,  10654827804,  17343084960,  10654827804,   2344429008,   152031492,    1679616;
  9: 10077696, 1418671908, 32237998380, 216088713252, 535926913164, 535926913164, 216088713252, 32237998380, 1418671908, 10077696;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+(3*n+3)*x)*P(n-1,x) + (3*x-3*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[(6 + (3*n + 3)*x)*P[n - 1, x] + (3*x - 3*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] = (6 + (3*n + 3)*x)*v[i - 1] + (3*x - 3*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-5ba7ec4213cfb517
  • 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-5ba7ec4213cfb517
  • 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-5ba7ec4213cfb517
  • 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-5ba7ec4213cfb517

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 3 proof steps

Same fingerprint (2)

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