A039755 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 3\,x + 2\,n x\right)\,P_{n-1}(x) + \left(-2\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	1							2
3	1	4	1						6
4	1	9	13	1					24
5	1	16	58	40	1				116
6	1	25	170	330	121	1			648
7	1	36	395	1520	1771	364	1		4088
8	1	49	791	5075	12411	9219	1093	1	28640

Show all 15 stored rows

Generated b-files: A007405 (row sums), A039755 (reversed triangle), A039756 ?

Computed in your browser ?

Technical details

Node id: n-f08d4bb0872e0b9e · revision 1 · JSON
Definition source: derived:reversal
Fingerprint: f6-c67245fc3f808c35

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 ?

A007405: 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 ?
A039755: reversed triangle, 56 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 ?
A039756: 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 ?

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,  4,    1;
   4: 1,  9,   13,     1;
   5: 1, 16,   58,    40,      1;
   6: 1, 25,  170,   330,    121,      1;
   7: 1, 36,  395,  1520,   1771,    364,      1;
   8: 1, 49,  791,  5075,  12411,   9219,   1093,      1;
   9: 1, 64, 1428, 13776,  58086,  96096,  47188,   3280,    1;
  10: 1, 81, 2388, 32340, 209622, 618870, 719860, 239220, 9841, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-3)*x)*P(n-1,x) - 2*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 - 3)*x)*P[n - 1, x] - 2*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {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 - 3)*x)*v[i - 1] - 2*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A039755: reversed triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    4,      1;
  3: 1,   13,      9,      1;
  4: 1,   40,     58,     16,      1;
  5: 1,  121,    330,    170,     25,      1;
  6: 1,  364,   1771,   1520,    395,     36,     1;
  7: 1, 1093,   9219,  12411,   5075,    791,    49,    1;
  8: 1, 3280,  47188,  96096,  58086,  13776,  1428,   64,  1;
  9: 1, 9841, 239220, 719860, 618870, 209622, 32340, 2388, 81, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(2*n-1)*x)*P(n-1,x) - 2*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 - 1)*x)*P[n - 1, x] - 2*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 - 1)*x)*v[i - 1] - 2*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.

A039756: triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,    1;
  3: 1,  9,   13,     1;
  4: 1, 16,   58,    40,      1;
  5: 1, 25,  170,   330,    121,      1;
  6: 1, 36,  395,  1520,   1771,    364,      1;
  7: 1, 49,  791,  5075,  12411,   9219,   1093,      1;
  8: 1, 64, 1428, 13776,  58086,  96096,  47188,   3280,    1;
  9: 1, 81, 2388, 32340, 209622, 618870, 719860, 239220, 9841, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-1)*x)*P(n-1,x) - 2*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 - 1)*x)*P[n - 1, x] - 2*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] = (1 + (2*n - 1)*x)*v[i - 1] - 2*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 >= 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-70d96bc14ba32af8
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-70d96bc14ba32af8
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-70d96bc14ba32af8

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 0 e-70d96bc14ba32af8

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle of B-analogs of Stirling numbers of the second kind.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-c873e8cf4f19d199 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (3)

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