Real-rooted atlas

A367270: Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).

A367270 nonnegative coefficients

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

Recurrence from OEIS A367270
\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(2 + 4\,x - 8\,n - 8\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 + 4\,x - 2\,x^{2} + 5\,n - 10\,n x + 5\,n x^{2} - 2\,n^{2} + 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2123
316310
411218435
512060405126
6130150200756462
714231570052512671716
815658819602450117619686435
917210084704882070562352288924310
1019016201008026460317521764043204051092378
111110247519800693001164249702039600742555011352716
1211323630363001633503659044268882613608167512100726121352078
13115651486292035392510193041585584135907263706515730018876936135200300
1411827098104104715715257657451531485889312386486114314302862862839211831420058300

Generated b-files: A001700 (row sums), A088218 (row sums), A103371 (reversed triangle), A132813 ?

Computed in your browser ?

Technical details
Node id
n-b1b19325e0094118 · revision 1 · JSON
Definition source
oeis_catalog:A367270
Fingerprint
f6-4b23e46a501a23b5

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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): 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: 1;
  2: 1,  2;
  3: 1,  6,    3;
  4: 1, 12,   18,    4;
  5: 1, 20,   60,   40,    5;
  6: 1, 30,  150,  200,   75,    6;
  7: 1, 42,  315,  700,  525,  126,    7;
  8: 1, 56,  588, 1960, 2450, 1176,  196,   8;
  9: 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9;
%F Formula
The row polynomials satisfy (2*n^2-3*n)*P(n,x) = (4*n^2-8*n+2+(4*n^2-8*n+4)*x)*P(n-1,x) - (2*n^2-5*n+2-(4*n^2-10*n+4)*x+(2*n^2-5*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 8*n + 2 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] - (2*n^2 - 5*n + 2 - (4*n^2 - 10*n + 4)*x + (2*n^2 - 5*n + 2)*x^2)*P[n - 2, x])/(2*n^2 - 3*n)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 8*n + 2 + (4*n^2 - 8*n + 4)*x)*v[i - 1] - (2*n^2 - 5*n + 2 - (4*n^2 - 10*n + 4)*x + (2*n^2 - 5*n + 2)*x^2)*v[i - 2])/(2*n^2 - 3*n)); 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.

A103371: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   6,    1;
  3:  4,  18,   12,     1;
  4:  5,  40,   60,    20,     1;
  5:  6,  75,  200,   150,    30,     1;
  6:  7, 126,  525,   700,   315,    42,     1;
  7:  8, 196, 1176,  2450,  1960,   588,    56,    1;
  8:  9, 288, 2352,  7056,  8820,  4704,  1008,   72,  1;
  9: 10, 405, 4320, 17640, 31752, 26460, 10080, 1620, 90, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n^2+n-1)*P(n,x) = (4*n^2-2+4*n^2*x)*P(n-1,x) - (2*n^2-n-1-(4*n^2-2*n-2)*x+(2*n^2-n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2 + 4*n^2*x)*P[n - 1, x] - (2*n^2 - n - 1 - (4*n^2 - 2*n - 2)*x + (2*n^2 - n - 1)*x^2)*P[n - 2, x])/(2*n^2 + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 2 + 4*n^2*x)*v[i - 1] - (2*n^2 - n - 1 - (4*n^2 - 2*n - 2)*x + (2*n^2 - n - 1)*x^2)*v[i - 2])/(2*n^2 + 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)

Generated with polytool c48629374597.

A132813: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  6,    3;
  3: 1, 12,   18,     4;
  4: 1, 20,   60,    40,     5;
  5: 1, 30,  150,   200,    75,     6;
  6: 1, 42,  315,   700,   525,   126,     7;
  7: 1, 56,  588,  1960,  2450,  1176,   196,    8;
  8: 1, 72, 1008,  4704,  8820,  7056,  2352,  288,   9;
  9: 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10;
%F Formula
The row polynomials satisfy (2*n^2+n-1)*P(n,x) = (4*n^2-2+4*n^2*x)*P(n-1,x) - (2*n^2-n-1-(4*n^2-2*n-2)*x+(2*n^2-n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2 + 4*n^2*x)*P[n - 1, x] - (2*n^2 - n - 1 - (4*n^2 - 2*n - 2)*x + (2*n^2 - n - 1)*x^2)*P[n - 2, x])/(2*n^2 + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 2 + 4*n^2*x)*v[i - 1] - (2*n^2 - n - 1 - (4*n^2 - 2*n - 2)*x + (2*n^2 - n - 1)*x^2)*v[i - 2])/(2*n^2 + 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-30d8fbe7ca664c09
  • 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-2b43df6c63e34146
  • 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-2b43df6c63e34146
  • 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-2b43df6c63e34146
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-2b43df6c63e34146

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 10 proof steps

Same fingerprint (3)

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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023