A056939: Array read by antidiagonals: number of antichains (or order ideals) in the poset 3mn or plane partitions with rows <= m, columns <= n and entries <= 3.

A108838 → ⊙ C(n+1,k+1) (shift +1)

A056939 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 A056939

\(\left(12 + 7\,n + n^{2}\right)\,P_{n}(x) = \left(12 + 9\,n + 12\,n x - n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(12\,x - 12\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x) + \left(-2\,n - 4\,n x - 2\,n x^{2} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	4	1						6
3	1	10	10	1					22
4	1	20	50	20	1				92
5	1	35	175	175	35	1			422
6	1	56	490	980	490	56	1		2074
7	1	84	1176	4116	4116	1176	84	1	10754

Show all 15 stored rows

Generated b-files: A056939, A056939 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(\left(12 + 7\,n + n^{2}\right)\,P_{n}(x) = \left(12 + 7\,n + 10\,n x + n^{2} + 10\,n^{2} x\right)\,P_{n-1}(x) + \left(24\,x + 3\,n x - 15\,n x^{2}\right)\,P_{n-1}'(x) + \left(6\,x^{2} + 6\,x^{3}\right)\,P_{n-1}''(x)\)

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

Technical details

Node id: n-047228a38014e6d4 · revision 2 · JSON
Definition source: oeis_catalog:A056939
derived:reversal (equal by e-77f09a180af02463)
Fingerprint: f6-509399d8a74ba170

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 ?

A056939: 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 ?
A001181: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A056939: 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 ?

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,   4,    1;
  3: 1,  10,   10,     1;
  4: 1,  20,   50,    20,      1;
  5: 1,  35,  175,   175,     35,      1;
  6: 1,  56,  490,   980,    490,     56,     1;
  7: 1,  84, 1176,  4116,   4116,   1176,    84,    1;
  8: 1, 120, 2520, 14112,  24696,  14112,  2520,  120,   1;
  9: 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1;

%F Formula

The row polynomials satisfy (n^2+7*n+12)*P(n,x) = -(n^2-9*n-12-(8*n^2+12*n)*x)*P(n-1,x) + ((9*n+12)*x-(9*n+12)*x^2)*d/dx P(n-1,x) + (2*n^2-2*n+(4*n^2-4*n)*x+(2*n^2-2*n)*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 - 9*n - 12 - (8*n^2 + 12*n)*x)*P[n - 1, x] + ((9*n + 12)*x - (9*n + 12)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*P[n - 2, x])/(n^2 + 7*n + 12)]];
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 - 9*n - 12 - (8*n^2 + 12*n)*x)*v[i - 1] + ((9*n + 12)*x - (9*n + 12)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*v[i - 2])/(n^2 + 7*n + 12)); 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.

A056939: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,    1;
  3: 1,  10,   10,     1;
  4: 1,  20,   50,    20,      1;
  5: 1,  35,  175,   175,     35,      1;
  6: 1,  56,  490,   980,    490,     56,     1;
  7: 1,  84, 1176,  4116,   4116,   1176,    84,    1;
  8: 1, 120, 2520, 14112,  24696,  14112,  2520,  120,   1;
  9: 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1;

%F Formula

The row polynomials satisfy (n^2+7*n+12)*P(n,x) = -(n^2-9*n-12-(8*n^2+12*n)*x)*P(n-1,x) + ((9*n+12)*x-(9*n+12)*x^2)*d/dx P(n-1,x) + (2*n^2-2*n+(4*n^2-4*n)*x+(2*n^2-2*n)*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 - 9*n - 12 - (8*n^2 + 12*n)*x)*P[n - 1, x] + ((9*n + 12)*x - (9*n + 12)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*P[n - 2, x])/(n^2 + 7*n + 12)]];
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 - 9*n - 12 - (8*n^2 + 12*n)*x)*v[i - 1] + ((9*n + 12)*x - (9*n + 12)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*v[i - 2])/(n^2 + 7*n + 12)); 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.

A056939: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,    1;
  3: 1,  10,   10,     1;
  4: 1,  20,   50,    20,      1;
  5: 1,  35,  175,   175,     35,      1;
  6: 1,  56,  490,   980,    490,     56,     1;
  7: 1,  84, 1176,  4116,   4116,   1176,    84,    1;
  8: 1, 120, 2520, 14112,  24696,  14112,  2520,  120,   1;
  9: 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2+7*n+12)*P(n,x) = -(n^2-9*n-12-(8*n^2+12*n)*x)*P(n-1,x) + ((9*n+12)*x-(9*n+12)*x^2)*d/dx P(n-1,x) + (2*n^2-2*n+(4*n^2-4*n)*x+(2*n^2-2*n)*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 - 9*n - 12 - (8*n^2 + 12*n)*x)*P[n - 1, x] + ((9*n + 12)*x - (9*n + 12)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*P[n - 2, x])/(n^2 + 7*n + 12)]];
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 - 9*n - 12 - (8*n^2 + 12*n)*x)*v[i - 1] + ((9*n + 12)*x - (9*n + 12)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*v[i - 2])/(n^2 + 7*n + 12)); 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-548c39a476f786f5, e-97ff2ea1491caf9e
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-548c39a476f786f5, e-97ff2ea1491caf9e
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-548c39a476f786f5, e-97ff2ea1491caf9e

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

definition_equality e-77f09a180af02463 root: theorem check

Ore-algebra identity of two definitions polytool ?
preserver_image hadamard_multiplier, shift 1 e-548c39a476f786f5

Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of Dyck paths counted by number of long interior inclines.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-a4549745e8cd4154 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
preserver_image reversal, shift 0 e-97ff2ea1491caf9e

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - ↵ preserver_image hadamard_multiplier, shift 1 e-548c39a476f786f5 (expanded above)

Used in 7 proof steps

e-23b30ca6e404c664 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Array read by antidiagonals: number of antichains (or order ideals) i…
e-0c759266bd791ca5 preserver_image (affine, shift 0) → shift_x_plus_one of Array read by antidiagonals: number of antichains (or order ideals) i…
e-5336a01b25274fbd preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Array read by antidiagonals: number of antichains (or order ideals…
e-97ff2ea1491caf9e preserver_image (reversal, shift 0) → Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or…
e-a929acd24243c7cc preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Array read by antidiagonals: number of antichains (or order i…
e-d09a0797aedc5af0 preserver_image (derivative_polynomial, shift 0) → derivative of Array read by antidiagonals: number of antichains (or order ideals) in the …
e-f6a16136bb4575f0 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in …

Related sequences ?

Same fingerprint (1)

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

A108838 → ⊙ C(n+2,k+1) (shift +2)

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