A288874: Row reversed version of triangle A201637 (second-order Eulerian triangle).

A201637 → reverse (shift +1)

A288874 nonnegative coefficients

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

Recurrence from OEIS A288874

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	1							1
2	0	2	1						3
3	0	6	8	1					15
4	0	24	58	22	1				105
5	0	120	444	328	52	1			945
6	0	720	3708	4400	1452	114	1		10395
7	0	5040	33984	58140	32120	5610	240	1	135135

Show all 15 stored rows

Generated b-files: A001147 (row sums), A008517 (reversed triangle), A112007, A201637 (reversed triangle), A288874 ?

Computed in your browser ?

Technical details

Node id: n-387e148a64b95986 · revision 1 · JSON
Definition source: oeis_catalog:A288874
Fingerprint: f6-36df5d4b0f5901bf

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: 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 ?

A112007: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A201637: 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 = 0 ?
A008517: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A001147: 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 ?
A288874: 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 ?

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,      2,       1;
  3: 0,      6,       8,        1;
  4: 0,     24,      58,       22,        1;
  5: 0,    120,     444,      328,       52,       1;
  6: 0,    720,    3708,     4400,     1452,     114,       1;
  7: 0,   5040,   33984,    58140,    32120,    5610,     240,     1;
  8: 0,  40320,  341136,   785304,   644020,  195800,   19950,   494,    1;
  9: 0, 362880, 3733920, 11026296, 12440064, 5765500, 1062500, 67260, 1004, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n-1+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 - 1 + 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 - 1 + 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.

A008517: reversed triangle, offset 1, from the row P₁, low zeros dropped

%e Example

Triangle begins:
   1: 1;
   2: 1,    2;
   3: 1,    8,      6;
   4: 1,   22,     58,      24;
   5: 1,   52,    328,     444,      120;
   6: 1,  114,   1452,    4400,     3708,       720;
   7: 1,  240,   5610,   32120,    58140,     33984,      5040;
   8: 1,  494,  19950,  195800,   644020,    785304,    341136,     40320;
   9: 1, 1004,  67260, 1062500,  5765500,  12440064,  11026296,   3733920,   362880;
  10: 1, 2026, 218848, 5326160, 44765000, 155357384, 238904904, 162186912, 44339040, 3628800;

%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-1+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 - 1 + 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 - 1 + 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.

A112007: triangle, offset 0, from the row P₁, low zeros dropped

%e Example

Triangle begins:
  0:       1;
  1:       2,        1;
  2:       6,        8,         1;
  3:      24,       58,        22,         1;
  4:     120,      444,       328,        52,         1;
  5:     720,     3708,      4400,      1452,       114,        1;
  6:    5040,    33984,     58140,     32120,      5610,      240,       1;
  7:   40320,   341136,    785304,    644020,    195800,    19950,     494,      1;
  8:  362880,  3733920,  11026296,  12440064,   5765500,  1062500,   67260,   1004,    1;
  9: 3628800, 44339040, 162186912, 238904904, 155357384, 44765000, 5326160, 218848, 2026, 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+(n+1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = x.

%t Mathematica

P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[(n + (n + 1)*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 + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = (n + (n + 1)*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 + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A201637: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,    0;
  2: 1,    2,     0;
  3: 1,    8,     6,       0;
  4: 1,   22,    58,      24,       0;
  5: 1,   52,   328,     444,     120,        0;
  6: 1,  114,  1452,    4400,    3708,      720,        0;
  7: 1,  240,  5610,   32120,   58140,    33984,     5040,       0;
  8: 1,  494, 19950,  195800,  644020,   785304,   341136,   40320,      0;
  9: 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880, 0;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n-1+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 - 1 + 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 - 1 + 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.

A288874: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 0,      1;
  2: 0,      2,       1;
  3: 0,      6,       8,        1;
  4: 0,     24,      58,       22,        1;
  5: 0,    120,     444,      328,       52,       1;
  6: 0,    720,    3708,     4400,     1452,     114,       1;
  7: 0,   5040,   33984,    58140,    32120,    5610,     240,     1;
  8: 0,  40320,  341136,   785304,   644020,  195800,   19950,   494,    1;
  9: 0, 362880, 3733920, 11026296, 12440064, 5765500, 1062500, 67260, 1004, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n-1+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 - 1 + 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 - 1 + 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.

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-e86572942f776a07
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-e86572942f776a07
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-e86572942f776a07

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-e86572942f776a07

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle of second-order 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-d7adf0401ce51e02 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (5)

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