A008288 → 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

\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(13 + 107\,x + 347\,x^{2} + 577\,x^{3} + 524\,x^{4} + 248\,x^{5} + 48\,x^{6} - 12\,n - 91\,n x - 285\,n x^{2} - 469\,n x^{3} - 427\,n x^{4} - 204\,n x^{5} - 40\,n x^{6} + 3\,n^{2} + 20\,n^{2} x + 58\,n^{2} x^{2} + 92\,n^{2} x^{3} + 83\,n^{2} x^{4} + 40\,n^{2} x^{5} + 8\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(2 + 39\,x + 212\,x^{2} + 546\,x^{3} + 774\,x^{4} + 623\,x^{5} + 268\,x^{6} + 48\,x^{7} - n - 14\,n x - 70\,n x^{2} - 176\,n x^{3} - 249\,n x^{4} - 202\,n x^{5} - 88\,n x^{6} - 16\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 13\,x^{2} + 58\,x^{3} + 130\,x^{4} + 165\,x^{5} + 121\,x^{6} + 48\,x^{7} + 8\,x^{8}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	2							3
3	1	5	5						11
4	1	8	18	12					39
5	1	11	40	58	29				139
6	1	14	71	164	175	70			495
7	1	17	111	357	601	507	169		1763
8	1	20	160	664	1550	2048	1428	408	6279

Show all 15 stored rows

Generated b-files: A007482 (row sums), A209695 ?

Computed in your browser ?

Technical details

Node id: n-1e3c6c28ffc91fd2 · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-f8b12148df73b33e

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 ?

A007482: 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 ?
A209695: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,   5;
   4: 1,  8,  18,   12;
   5: 1, 11,  40,   58,   29;
   6: 1, 14,  71,  164,  175,    70;
   7: 1, 17, 111,  357,  601,   507,   169;
   8: 1, 20, 160,  664, 1550,  2048,  1428,   408;
   9: 1, 23, 218, 1112, 3346,  6106,  6632,  3940,   985;
  10: 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378;

%F Formula

The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (3*n^2-12*n+13+(20*n^2-91*n+107)*x+(58*n^2-285*n+347)*x^2+(92*n^2-469*n+577)*x^3+(83*n^2-427*n+524)*x^4+(40*n^2-204*n+248)*x^5+(8*n^2-40*n+48)*x^6)*P(n-1,x) - (n-2+(14*n-39)*x+(70*n-212)*x^2+(176*n-546)*x^3+(249*n-774)*x^4+(202*n-623)*x^5+(88*n-268)*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 >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 12*n + 13 + (20*n^2 - 91*n + 107)*x + (58*n^2 - 285*n + 347)*x^2 + (92*n^2 - 469*n + 577)*x^3 + (83*n^2 - 427*n + 524)*x^4 + (40*n^2 - 204*n + 248)*x^5 + (8*n^2 - 40*n + 48)*x^6)*P[n - 1, x] - (n - 2 + (14*n - 39)*x + (70*n - 212)*x^2 + (176*n - 546)*x^3 + (249*n - 774)*x^4 + (202*n - 623)*x^5 + (88*n - 268)*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 - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];
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] = ((3*n^2 - 12*n + 13 + (20*n^2 - 91*n + 107)*x + (58*n^2 - 285*n + 347)*x^2 + (92*n^2 - 469*n + 577)*x^3 + (83*n^2 - 427*n + 524)*x^4 + (40*n^2 - 204*n + 248)*x^5 + (8*n^2 - 40*n + 48)*x^6)*v[i - 1] - (n - 2 + (14*n - 39)*x + (70*n - 212)*x^2 + (176*n - 546)*x^3 + (249*n - 774)*x^4 + (202*n - 623)*x^5 + (88*n - 268)*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 - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*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.

A209695: triangle, offset 1, from the row P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,   5;
   4: 1,  8,  18,   12;
   5: 1, 11,  40,   58,   29;
   6: 1, 14,  71,  164,  175,    70;
   7: 1, 17, 111,  357,  601,   507,   169;
   8: 1, 20, 160,  664, 1550,  2048,  1428,   408;
   9: 1, 23, 218, 1112, 3346,  6106,  6632,  3940,   985;
  10: 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378;

%F Formula

The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (3*n^2-12*n+13+(20*n^2-91*n+107)*x+(58*n^2-285*n+347)*x^2+(92*n^2-469*n+577)*x^3+(83*n^2-427*n+524)*x^4+(40*n^2-204*n+248)*x^5+(8*n^2-40*n+48)*x^6)*P(n-1,x) - (n-2+(14*n-39)*x+(70*n-212)*x^2+(176*n-546)*x^3+(249*n-774)*x^4+(202*n-623)*x^5+(88*n-268)*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 >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 12*n + 13 + (20*n^2 - 91*n + 107)*x + (58*n^2 - 285*n + 347)*x^2 + (92*n^2 - 469*n + 577)*x^3 + (83*n^2 - 427*n + 524)*x^4 + (40*n^2 - 204*n + 248)*x^5 + (8*n^2 - 40*n + 48)*x^6)*P[n - 1, x] - (n - 2 + (14*n - 39)*x + (70*n - 212)*x^2 + (176*n - 546)*x^3 + (249*n - 774)*x^4 + (202*n - 623)*x^5 + (88*n - 268)*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 - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*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; for(n = 2, N, my(i = n); v[i] = ((3*n^2 - 12*n + 13 + (20*n^2 - 91*n + 107)*x + (58*n^2 - 285*n + 347)*x^2 + (92*n^2 - 469*n + 577)*x^3 + (83*n^2 - 427*n + 524)*x^4 + (40*n^2 - 204*n + 248)*x^5 + (8*n^2 - 40*n + 48)*x^6)*v[i - 1] - (n - 2 + (14*n - 39)*x + (70*n - 212)*x^2 + (176*n - 546)*x^3 + (249*n - 774)*x^4 + (202*n - 623)*x^5 + (88*n - 268)*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 - 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 - #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-f285ab060a211a2e
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-f285ab060a211a2e
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-f285ab060a211a2e

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-f285ab060a211a2e

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
  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-b176436450506e78 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-73f1cbc14f4fad08
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
    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-b176436450506e78 (expanded above)
    
    basis_map A008288 e-796ce5140a0b4b04
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    basis_map A008288 e-b8e0a0fa3ba7bf57
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
  - ↵ basis_map A008288 e-796ce5140a0b4b04 (expanded above)
  - ↵ basis_map A008288 e-b8e0a0fa3ba7bf57 (expanded above)

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