A035607 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(4 + 54\,x + 226\,x^{2} + 442\,x^{3} + 450\,x^{4} + 232\,x^{5} + 48\,x^{6} - 8\,n - 67\,n x - 229\,n x^{2} - 405\,n x^{3} - 391\,n x^{4} - 196\,n x^{5} - 40\,n x^{6} + 3\,n^{2} + 20\,n^{2} x + 58\,n^{2} x^{2} + 92\,n^{2} x^{3} + 83\,n^{2} x^{4} + 40\,n^{2} x^{5} + 8\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(1 + 27\,x + 166\,x^{2} + 462\,x^{3} + 693\,x^{4} + 583\,x^{5} + 260\,x^{6} + 48\,x^{7} - n - 14\,n x - 70\,n x^{2} - 176\,n x^{3} - 249\,n x^{4} - 202\,n x^{5} - 88\,n x^{6} - 16\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 13\,x^{2} + 58\,x^{3} + 130\,x^{4} + 165\,x^{5} + 121\,x^{6} + 48\,x^{7} + 8\,x^{8}\right)\,P_{n-1}''(x)\)

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

\(P_{2}(x) = 1 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	3							4
3	1	6	7						14
4	1	9	23	17					50
5	1	12	48	76	41				178
6	1	15	82	204	233	99			634
7	1	18	125	428	765	682	239		2258
8	1	21	177	775	1907	2649	1935	577	8042

Show all 15 stored rows

Generated b-files: A054458 (reversed triangle), A055099 (row sums), A209696 ?

Computed in your browser ?

Technical details

Node id: n-7a1706417cb89af7 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-b42d277a4b716fed

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 ?

A055099: 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 ?
A209696: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 1 ?
A054458: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · 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,  3;
   3: 1,  6,   7;
   4: 1,  9,  23,   17;
   5: 1, 12,  48,   76,   41;
   6: 1, 15,  82,  204,  233,    99;
   7: 1, 18, 125,  428,  765,   682,   239;
   8: 1, 21, 177,  775, 1907,  2649,  1935,   577;
   9: 1, 24, 238, 1272, 4010,  7656,  8680,  5368,  1393;
  10: 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363;

%F Formula

The row polynomials satisfy (n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n,x) = (3*n^2-8*n+4+(20*n^2-67*n+54)*x+(58*n^2-229*n+226)*x^2+(92*n^2-405*n+442)*x^3+(83*n^2-391*n+450)*x^4+(40*n^2-196*n+232)*x^5+(8*n^2-40*n+48)*x^6)*P(n-1,x) - (n-1+(14*n-27)*x+(70*n-166)*x^2+(176*n-462)*x^3+(249*n-693)*x^4+(202*n-583)*x^5+(88*n-260)*x^6+(16*n-48)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 8*n + 4 + (20*n^2 - 67*n + 54)*x + (58*n^2 - 229*n + 226)*x^2 + (92*n^2 - 405*n + 442)*x^3 + (83*n^2 - 391*n + 450)*x^4 + (40*n^2 - 196*n + 232)*x^5 + (8*n^2 - 40*n + 48)*x^6)*P[n - 1, x] - (n - 1 + (14*n - 27)*x + (70*n - 166)*x^2 + (176*n - 462)*x^3 + (249*n - 693)*x^4 + (202*n - 583)*x^5 + (88*n - 260)*x^6 + (16*n - 48)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)]];
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 + 3*x; for(n = 3, N, my(i = n); v[i] = ((3*n^2 - 8*n + 4 + (20*n^2 - 67*n + 54)*x + (58*n^2 - 229*n + 226)*x^2 + (92*n^2 - 405*n + 442)*x^3 + (83*n^2 - 391*n + 450)*x^4 + (40*n^2 - 196*n + 232)*x^5 + (8*n^2 - 40*n + 48)*x^6)*v[i - 1] - (n - 1 + (14*n - 27)*x + (70*n - 166)*x^2 + (176*n - 462)*x^3 + (249*n - 693)*x^4 + (202*n - 583)*x^5 + (88*n - 260)*x^6 + (16*n - 48)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)); 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.

A054458: reversed triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0:    1;
  1:    3,     1;
  2:    7,     6,     1;
  3:   17,    23,     9,     1;
  4:   41,    76,    48,    12,     1;
  5:   99,   233,   204,    82,    15,    1;
  6:  239,   682,   765,   428,   125,   18,    1;
  7:  577,  1935,  2649,  1907,   775,  177,   21,   1;
  8: 1393,  5368,  8680,  7656,  4010, 1272,  238,  24,  1;
  9: 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (3*n^2-2*n-1+(20*n^2-27*n+7)*x+(58*n^2-113*n+55)*x^2+(92*n^2-221*n+129)*x^3+(83*n^2-225*n+142)*x^4+(40*n^2-116*n+76)*x^5+(8*n^2-24*n+16)*x^6)*P(n-1,x) - (n+(14*n-13)*x+(70*n-96)*x^2+(176*n-286)*x^3+(249*n-444)*x^4+(202*n-381)*x^5+(88*n-172)*x^6+(16*n-32)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 2*n - 1 + (20*n^2 - 27*n + 7)*x + (58*n^2 - 113*n + 55)*x^2 + (92*n^2 - 221*n + 129)*x^3 + (83*n^2 - 225*n + 142)*x^4 + (40*n^2 - 116*n + 76)*x^5 + (8*n^2 - 24*n + 16)*x^6)*P[n - 1, x] - (n + (14*n - 13)*x + (70*n - 96)*x^2 + (176*n - 286)*x^3 + (249*n - 444)*x^4 + (202*n - 381)*x^5 + (88*n - 172)*x^6 + (16*n - 32)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];
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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^2 - 2*n - 1 + (20*n^2 - 27*n + 7)*x + (58*n^2 - 113*n + 55)*x^2 + (92*n^2 - 221*n + 129)*x^3 + (83*n^2 - 225*n + 142)*x^4 + (40*n^2 - 116*n + 76)*x^5 + (8*n^2 - 24*n + 16)*x^6)*v[i - 1] - (n + (14*n - 13)*x + (70*n - 96)*x^2 + (176*n - 286)*x^3 + (249*n - 444)*x^4 + (202*n - 381)*x^5 + (88*n - 172)*x^6 + (16*n - 32)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)); 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.

A209696: triangle, offset 1, from the row P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,  3;
   3: 1,  6,   7;
   4: 1,  9,  23,   17;
   5: 1, 12,  48,   76,   41;
   6: 1, 15,  82,  204,  233,    99;
   7: 1, 18, 125,  428,  765,   682,   239;
   8: 1, 21, 177,  775, 1907,  2649,  1935,   577;
   9: 1, 24, 238, 1272, 4010,  7656,  8680,  5368,  1393;
  10: 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363;

%F Formula

The row polynomials satisfy (n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n,x) = (3*n^2-8*n+4+(20*n^2-67*n+54)*x+(58*n^2-229*n+226)*x^2+(92*n^2-405*n+442)*x^3+(83*n^2-391*n+450)*x^4+(40*n^2-196*n+232)*x^5+(8*n^2-40*n+48)*x^6)*P(n-1,x) - (n-1+(14*n-27)*x+(70*n-166)*x^2+(176*n-462)*x^3+(249*n-693)*x^4+(202*n-583)*x^5+(88*n-260)*x^6+(16*n-48)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 8*n + 4 + (20*n^2 - 67*n + 54)*x + (58*n^2 - 229*n + 226)*x^2 + (92*n^2 - 405*n + 442)*x^3 + (83*n^2 - 391*n + 450)*x^4 + (40*n^2 - 196*n + 232)*x^5 + (8*n^2 - 40*n + 48)*x^6)*P[n - 1, x] - (n - 1 + (14*n - 27)*x + (70*n - 166)*x^2 + (176*n - 462)*x^3 + (249*n - 693)*x^4 + (202*n - 583)*x^5 + (88*n - 260)*x^6 + (16*n - 48)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = ((3*n^2 - 8*n + 4 + (20*n^2 - 67*n + 54)*x + (58*n^2 - 229*n + 226)*x^2 + (92*n^2 - 405*n + 442)*x^3 + (83*n^2 - 391*n + 450)*x^4 + (40*n^2 - 196*n + 232)*x^5 + (8*n^2 - 40*n + 48)*x^6)*v[i - 1] - (n - 1 + (14*n - 27)*x + (70*n - 166)*x^2 + (176*n - 462)*x^3 + (249*n - 693)*x^4 + (202*n - 583)*x^5 + (88*n - 260)*x^6 + (16*n - 48)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)); 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.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-4b3874b4d357a754
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-4b3874b4d357a754
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-4b3874b4d357a754

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 mobius, shift 0 e-4b3874b4d357a754

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-f5199491b1877911 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

Same fingerprint (1)

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

A113413 → basis A007318

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