A123125: Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.

A173018 → reverse (shift +1)

A123125 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A123125

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

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	1							1
2	0	1	1						2
3	0	1	4	1					6
4	0	1	11	11	1				24
5	0	1	26	66	26	1			120
6	0	1	57	302	302	57	1		720
7	0	1	120	1191	2416	1191	120	1	5040

Show all 15 stored rows

Generated b-files: A000142 (row sums), A008292, A008292 (reversed triangle), A123125, A173018 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-e24b36ff0794dfbd · revision 1 · JSON
Definition source: oeis_catalog:A123125
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 ?

A159333: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
A124355: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A008292: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A370360: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A123125: triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A000142: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A358185: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
A173018: reversed triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A008292: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A104150: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A354074: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A338112: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 0, 1;
  2: 0, 1,   1;
  3: 0, 1,   4,     1;
  4: 0, 1,  11,    11,     1;
  5: 0, 1,  26,    66,    26,      1;
  6: 0, 1,  57,   302,   302,     57,     1;
  7: 0, 1, 120,  1191,  2416,   1191,   120,     1;
  8: 0, 1, 247,  4293, 15619,  15619,  4293,   247,   1;
  9: 0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1;

%F Formula

The row polynomials satisfy P(n,x) = n*x*P(n-1,x) + (x-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[n*x*P[n - 1, x] + (x - 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] = n*x*v[i - 1] + (x - 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.

A008292: triangle, offset 1, from the row P₁, low zeros dropped

%e Example

Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    4,     1;
   4: 1,   11,    11,      1;
   5: 1,   26,    66,     26,       1;
   6: 1,   57,   302,    302,      57,       1;
   7: 1,  120,  1191,   2416,    1191,     120,      1;
   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;
   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;
  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = n*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.

%t Mathematica

P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[n*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = n*x*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A008292: reversed triangle, offset 1, from the row P₁, low zeros dropped

%e Example

Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    4,     1;
   4: 1,   11,    11,      1;
   5: 1,   26,    66,     26,       1;
   6: 1,   57,   302,    302,      57,       1;
   7: 1,  120,  1191,   2416,    1191,     120,      1;
   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;
   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;
  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = n*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.

%t Mathematica

P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[n*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = n*x*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A123125: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,   1;
  3: 0, 1,   4,     1;
  4: 0, 1,  11,    11,     1;
  5: 0, 1,  26,    66,    26,      1;
  6: 0, 1,  57,   302,   302,     57,     1;
  7: 0, 1, 120,  1191,  2416,   1191,   120,     1;
  8: 0, 1, 247,  4293, 15619,  15619,  4293,   247,   1;
  9: 0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1;

%F Formula

The row polynomials satisfy P(n,x) = n*x*P(n-1,x) + (x-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[n*x*P[n - 1, x] + (x - 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] = n*x*v[i - 1] + (x - 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.

A173018: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,   0;
  2: 1,   1,     0;
  3: 1,   4,     1,     0;
  4: 1,  11,    11,     1,      0;
  5: 1,  26,    66,    26,      1,     0;
  6: 1,  57,   302,   302,     57,     1,     0;
  7: 1, 120,  1191,  2416,   1191,   120,     1,   0;
  8: 1, 247,  4293, 15619,  15619,  4293,   247,   1, 0;
  9: 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 0;

%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) + (x-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[n*x*P[n - 1, x] + (x - 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] = n*x*v[i - 1] + (x - 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-61bc78adfe1eb25a
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-61bc78adfe1eb25a
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-61bc78adfe1eb25a

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

preserver_image reversal, shift 1 e-61bc78adfe1eb25a

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-083b494dcfd2f637 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-ffc8a9fffd6e96cb root: theorem check
    
    Parent-row provers of polytool polytool ?

Used in 7 proof steps

e-52ca2f2e23126004 preserver_image (reversal, shift 0) → reversal of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
e-6aabb2282cf76ea5 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
e-a05b599234f2dfc5 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
e-afbdda6cf256c7ee preserver_image (affine, shift 0) → Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(…
e-bb4a78f786290f50 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
e-bf324d0a6ff5828b preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
e-e273b6c51200d112 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by row…

Related sequences ?

Same fingerprint (6)

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

References

[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