A181543: Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.

A008459 → ⊙ C(n,k)

A181543 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A181543

\(\left(n^{2}\right)\,P_{n}(x) = \left(-2 + 4\,x + 4\,n - 11\,n x - n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x) + \left(2 + 4\,x + 2\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	8	1						10
3	1	27	27	1					56
4	1	64	216	64	1				346
5	1	125	1000	1000	125	1			2252
6	1	216	3375	8000	3375	216	1		15184
7	1	343	9261	42875	42875	9261	343	1	104960

Show all 15 stored rows

Generated b-files: A000172 (row sums), A181543, A181543 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-fabb0a68e001a863 · revision 1 · JSON
Definition source: oeis_catalog:A181543
Fingerprint: f6-a35162fa3482274f

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A000172: 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 ?
A181543: reversed 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 ?
A181543: 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   8,     1;
  3: 1,  27,    27,      1;
  4: 1,  64,   216,     64,       1;
  5: 1, 125,  1000,   1000,     125,       1;
  6: 1, 216,  3375,   8000,    3375,     216,      1;
  7: 1, 343,  9261,  42875,   42875,    9261,    343,     1;
  8: 1, 512, 21952, 175616,  343000,  175616,  21952,   512,   1;
  9: 1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1;

%F Formula

The row polynomials satisfy n^2*P(n,x) = -(n^2-4*n+2-(8*n^2-11*n+4)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) + (2*n^2-4*n+2+(4*n^2-8*n+4)*x+(2*n^2-4*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*P[n - 2, x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*v[i - 2])/(n^2)); 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.

A181543: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   8,     1;
  3: 1,  27,    27,      1;
  4: 1,  64,   216,     64,       1;
  5: 1, 125,  1000,   1000,     125,       1;
  6: 1, 216,  3375,   8000,    3375,     216,      1;
  7: 1, 343,  9261,  42875,   42875,    9261,    343,     1;
  8: 1, 512, 21952, 175616,  343000,  175616,  21952,   512,   1;
  9: 1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1;

%F Formula

The row polynomials satisfy n^2*P(n,x) = -(n^2-4*n+2-(8*n^2-11*n+4)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) + (2*n^2-4*n+2+(4*n^2-8*n+4)*x+(2*n^2-4*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*P[n - 2, x])/(n^2)]];
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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*v[i - 2])/(n^2)); 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.

A181543: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   8,     1;
  3: 1,  27,    27,      1;
  4: 1,  64,   216,     64,       1;
  5: 1, 125,  1000,   1000,     125,       1;
  6: 1, 216,  3375,   8000,    3375,     216,      1;
  7: 1, 343,  9261,  42875,   42875,    9261,    343,     1;
  8: 1, 512, 21952, 175616,  343000,  175616,  21952,   512,   1;
  9: 1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where n^2*P(n,x) = -(n^2-4*n+2-(8*n^2-11*n+4)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) + (2*n^2-4*n+2+(4*n^2-8*n+4)*x+(2*n^2-4*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*P[n - 2, x])/(n^2)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*v[i - 2])/(n^2)); 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-835093125a0f64b6
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-835093125a0f64b6
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-835093125a0f64b6

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 hadamard_multiplier, shift 0 e-835093125a0f64b6

Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Square the entries of Pascal's triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-c6b04c4560617585 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-1906182487ade7a7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square the entries of Pascal's triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-c6b04c4560617585 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023