A113953 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-11\,n x + 5\,n x^{2} + n x^{3} - 2\,n^{2} + 7\,n^{2} x + n^{2} x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-48\,x + 16\,x^{2} + 10\,x^{3} + x^{4} - 24\,n + 66\,n x - 10\,n x^{2} - 3\,n x^{3} + 32\,n^{2} - 22\,n^{2} x - 4\,n^{2} x^{2} - 8\,n^{3}\right)\,P_{n-1}(x) + \left(-32\,x^{2} + 52\,x^{3} + 15\,x^{4} + x^{5} - 16\,n x + 54\,n x^{2} + 15\,n x^{3} + n x^{4} + 16\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	2	4	1						7
3	0	8	7	1					16
4	0	8	22	10	1				41
5	0	0	36	44	13	1			94
6	0	0	24	104	74	16	1		219
7	0	0	0	128	232	112	19	1	492

Show all 15 stored rows

Generated b-files: A073371 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-d1ec595c61033c43 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2450a51cfab3abfe

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 ?

A073371: 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 ?

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, 1;
  2: 2, 4,  1;
  3: 0, 8,  7,   1;
  4: 0, 8, 22,  10,   1;
  5: 0, 0, 36,  44,  13,    1;
  6: 0, 0, 24, 104,  74,   16,   1;
  7: 0, 0,  0, 128, 232,  112,  19,   1;
  8: 0, 0,  0,  64, 416,  440, 158,  22,  1;
  9: 0, 0,  0,   0, 400, 1040, 748, 212, 25, 1;

%F Formula

The row polynomials satisfy (2*n^3-2*n^2+(7*n^2-11*n)*x+(n^2+5*n)*x^2+n*x^3)*P(n,x) = -(8*n^3-32*n^2+24*n+(22*n^2-66*n+48)*x+(4*n^2+10*n-16)*x^2+(3*n-10)*x^3-x^4)*P(n-1,x) + ((16*n^2-16*n)*x+(2*n^2+54*n-32)*x^2+(15*n+52)*x^3+(n+15)*x^4+x^5)*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[Together[(-(8*n^3 - 32*n^2 + 24*n + (22*n^2 - 66*n + 48)*x + (4*n^2 + 10*n - 16)*x^2 + (3*n - 10)*x^3 - x^4)*P[n - 1, x] + ((16*n^2 - 16*n)*x + (2*n^2 + 54*n - 32)*x^2 + (15*n + 52)*x^3 + (n + 15)*x^4 + x^5)*D[P[n - 1, x], x])/(2*n^3 - 2*n^2 + (7*n^2 - 11*n)*x + (n^2 + 5*n)*x^2 + n*x^3)]];
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] = (-(8*n^3 - 32*n^2 + 24*n + (22*n^2 - 66*n + 48)*x + (4*n^2 + 10*n - 16)*x^2 + (3*n - 10)*x^3 - x^4)*v[i - 1] + ((16*n^2 - 16*n)*x + (2*n^2 + 54*n - 32)*x^2 + (15*n + 52)*x^3 + (n + 15)*x^4 + x^5)*deriv(v[i - 1], x))/(2*n^3 - 2*n^2 + (7*n^2 - 11*n)*x + (n^2 + 5*n)*x^2 + n*x^3)); 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.

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-5e8f0cb9241868fa
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-5e8f0cb9241868fa
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-5e8f0cb9241868fa

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 derivative_polynomial, shift 0 e-5e8f0cb9241868fa

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): A Jacobsthal triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A027467 e-0db30c1d149f8e9a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-0ffa0091fcab81ab
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    theorem_check ma_wang_interval e-2d5b16bf2955a407 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
  - basis_map A038303 e-1078e1fe9c978d23
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
    
    preserver_image affine, shift 0 e-ba5819a43d999a58
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
  - basis_map A038327 e-39068ae2f1d878f6
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-0ef3f7736abad08b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
  - preserver_image reversal, shift 0 e-5066005127b21217
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    gamma_polynomial scale 1, shift 0 e-250eeed7ad07d148
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang_derivative_lag e-1f8dc4b132ba6116 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
    
    preserver_image reversal, shift 0 e-8efc9aafb30c2198
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-1f8dc4b132ba6116 (expanded above)
    
    preserver_image reversal, shift 0 e-6b9d711da9b2e815
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ basis_map A027467 e-0db30c1d149f8e9a (expanded above)
    
    ↵ basis_map A038303 e-1078e1fe9c978d23 (expanded above)
    
    ↵ basis_map A038327 e-39068ae2f1d878f6 (expanded above)
    
    basis_map A027465 e-76bf35985f9d5eb4
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-5f9e6216c74dfca3
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038243 e-7b55c4767302dd2a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
    
    preserver_image affine, shift 0 e-0c61b86c2e25257b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A007318 e-89ed187733702f47
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-38de1ebb8f2e73b6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038255 e-9c1dd42dfa910ea7
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-4867241b67312c9e
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038207 e-a0ae771232cbba16
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
    
    preserver_image affine, shift 0 e-c345fe1d5254381d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038291 e-a2d6a4286b164d3d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
    
    preserver_image affine, shift 0 e-f1a9ca5f4baaa1dd
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038231 e-b4c854eb781c5167
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-b26782bfa4993a06
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038279 e-c41010d0d9363851
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-765910d978fe3ce9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A027466 e-de853701eca80777
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-2f0380ac472e9edc
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A113953 e-4e53ceff1d4735da
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    parent_provers paper_theorem e-7c3667820056c57c root: theorem check
    
    Parent-row provers of polytool polytool ?
  - ↵ basis_map A027465 e-76bf35985f9d5eb4 (expanded above)
  - ↵ basis_map A038243 e-7b55c4767302dd2a (expanded above)
  - ↵ basis_map A007318 e-89ed187733702f47 (expanded above)
  - ↵ basis_map A038255 e-9c1dd42dfa910ea7 (expanded above)
  - ↵ basis_map A038207 e-a0ae771232cbba16 (expanded above)
  - ↵ basis_map A038291 e-a2d6a4286b164d3d (expanded above)
  - ↵ basis_map A038231 e-b4c854eb781c5167 (expanded above)
  - ↵ basis_map A038279 e-c41010d0d9363851 (expanded above)
  - ↵ basis_map A027466 e-de853701eca80777 (expanded above)
  - ↵ basis_map A113953 e-4e53ceff1d4735da (expanded above)
  - ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	2	4	1						7
3	0	8	7	1					16
4	0	8	22	10	1				41
5	0	0	36	44	13	1			94
6	0	0	24	104	74	16	1		219
7	0	0	0	128	232	112	19	1	492

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	2	4	1						7
3	0	8	7	1					16
4	0	8	22	10	1				41
5	0	0	36	44	13	1			94
6	0	0	24	104	74	16	1		219
7	0	0	0	128	232	112	19	1	492

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	2	4	1						7
3	0	8	7	1					16
4	0	8	22	10	1				41
5	0	0	36	44	13	1			94
6	0	0	24	104	74	16	1		219
7	0	0	0	128	232	112	19	1	492