A008517: Second-order Eulerian triangle T(n,k), 1 <= k <= n.

A008517 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 A008517

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	8	6						15
3	1	22	58	24					105
4	1	52	328	444	120				945
5	1	114	1452	4400	3708	720			10395
6	1	240	5610	32120	58140	33984	5040		135135
7	1	494	19950	195800	644020	785304	341136	40320	2027025

Show all 15 stored rows

Generated b-files: A008517, A112007 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-d4c21302c908a2cf · revision 1 · JSON
Definition source: oeis_catalog:A008517
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 ?

A001147: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A112007: 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 = 0 ?
A008517: 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 ?

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

%F Formula

The row polynomials satisfy P(n,x) = (1+2*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[(1 + 2*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] = (1 + 2*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: triangle, offset 1, from the row P₀

%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

The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

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

%o PARI/GP

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

Generated with polytool c48629374597.

A112007: reversed triangle, offset 0, from the row P₀

%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 descending powers of x, where P(n,x) = (1+2*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[(1 + 2*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] = (1 + 2*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-83880dab33818290
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-83880dab33818290
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-83880dab33818290
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-83880dab33818290

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

theorem_check liu_wang e-83880dab33818290 root: theorem check

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 2 proof steps

e-5af5737c106529de obreschkoff_combination → obreschkoff_combination of Second-order Eulerian triangle T(n,k), 1 <= k <= n.
e-5af5737c106529de obreschkoff_combination → obreschkoff_combination of Second-order Eulerian triangle T(n,k), 1 <= k <= n.

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