Real-rooted atlas

A094816: Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.

A094816 nonnegative coefficients

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

Recurrence from OEIS A094816
\(P_{n}(x) = \left(-1 + x + n\right)\,P_{n-1}(x) + \left(2 - n\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31315
4186116
51242910165
618914575151326
714158145451602111957
81237252434179157530128113700

Show all 15 stored rows

Generated b-files: A000522 (row sums), A046716 (reversed triangle), A094816 ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(P_{n}(x) = \left(2 - n\right)\,P_{n-2}(x) + \left(-1 + x + n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)
Technical details
Node id
n-3272d5bc697b9cb4 · revision 2 · JSON
Definition source
oeis_catalog:A094816
derived:reversal (equal by e-20bfe0ff7ad51468)
Fingerprint
f6-7bb830754838abe6

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,      1;
   3: 1,      3,      1;
   4: 1,      8,      6,      1;
   5: 1,     24,     29,     10,      1;
   6: 1,     89,    145,     75,     15,     1;
   7: 1,    415,    814,    545,    160,    21,    1;
   8: 1,   2372,   5243,   4179,   1575,   301,   28,   1;
   9: 1,  16072,  38618,  34860,  15659,  3836,  518,  36,  1;
  10: 1, 125673, 321690, 318926, 163191, 47775, 8274, 834, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n-1+x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(n - 1 + x)*P[n - 1, x] - (n - 2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (n - 1 + x)*v[i - 1] - (n - 2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A046716: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   1;
  3: 1,  6,   8,    1;
  4: 1, 10,  29,   24,     1;
  5: 1, 15,  75,  145,    89,      1;
  6: 1, 21, 160,  545,   814,    415,      1;
  7: 1, 28, 301, 1575,  4179,   5243,   2372,      1;
  8: 1, 36, 518, 3836, 15659,  34860,  38618,  16072,      1;
  9: 1, 45, 834, 8274, 47775, 163191, 318926, 321690, 125673, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+x)*P(n-1,x) - (n-1)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(n + x)*P[n - 1, x] - (n - 1)*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + x)*v[i - 1] - (n - 1)*v[i - 2]); 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.

A094816: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,      1;
  2: 1,      3,      1;
  3: 1,      8,      6,      1;
  4: 1,     24,     29,     10,      1;
  5: 1,     89,    145,     75,     15,     1;
  6: 1,    415,    814,    545,    160,    21,    1;
  7: 1,   2372,   5243,   4179,   1575,   301,   28,   1;
  8: 1,  16072,  38618,  34860,  15659,  3836,  518,  36,  1;
  9: 1, 125673, 321690, 318926, 163191, 47775, 8274, 834, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+x)*P(n-1,x) - (n-1)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(n + x)*P[n - 1, x] - (n - 1)*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (n + x)*v[i - 1] - (n - 1)*v[i - 2]); 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 >= 1, 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-424a7fb0b4968342
  • 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-424a7fb0b4968342, e-503a12a637bb094e
  • Real-rooted. Every row P_n, n >= 1, 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-503a12a637bb094e
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-10ee31afdeccb0f4, e-503a12a637bb094e
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-10ee31afdeccb0f4, e-503a12a637bb094e
  • Real-rooted. Every row P_n, n >= 1, 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-10ee31afdeccb0f4

Proof chains

4 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 14 proof steps

Same fingerprint (1)

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