A060693 -id:A060693 - cp's OEIS frontend

A215287 Number of permutations of 0..floor((n*3-1)/2) on even squares of an n X 3 array such that each row and column of even squares is increasing.

1, 3, 10, 30, 140, 420, 2310, 6930, 42042, 126126, 816816, 2450448, 16628040, 49884120, 350574510, 1051723530, 7595781050, 22787343150, 168212023980, 504636071940, 3792416540640, 11377249621920, 86787993910800, 260363981732400, 2011383287449200

Offset: 1

R. H. Hardin, Aug 07 2012

nonn

Also Schröder paths of length n having floor(n/2) peaks. - Peter Luschny, Sep 30 2018

			Some solutions for n=5:
  0 x 4   0 x 5   1 x 3   0 x 1   0 x 3   1 x 4   0 x 2
  x 3 x   x 1 x   x 0 x   x 4 x   x 2 x   x 0 x   x 1 x
  1 x 5   2 x 6   2 x 5   2 x 3   1 x 6   2 x 5   3 x 5
  x 7 x   x 3 x   x 6 x   x 6 x   x 5 x   x 6 x   x 6 x
  2 x 6   4 x 7   4 x 7   5 x 7   4 x 7   3 x 7   4 x 7

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 3 of A215292.

Cf. A060854, A060693.

[(n-(n div 2)+1)*Factorial(2*n-(n div 2)) / (Factorial(n-(n div 2) +1)^2*Factorial((n div 2))): n in [1..30]]; // Vincenzo Librandi, Oct 01 2018

T := (n, k) -> (n-k+1)*(2*n-k)!/((n-k+1)!^2*k!):
a := n -> T(n, floor(n/2)): seq(a(n), n = 1..23); # Peter Luschny, Sep 30 2018

Table[(n - Floor[n/2] + 1) (2 n - Floor[n/2])! / ((n -Floor[n/2] + 1)!^2 Floor[n/2]!), {n, 1, 30}] (* Vincenzo Librandi, Oct 01 2018 *)

f3 = floor((n+1)/2); f4 = floor(n/2);
a(n) = A060854(2,f3)*A060854(1,f4)*binomial(2*f3+1*f4,2*f3).
a(n) = (n - f + 1)*(2*n - f)! / ((n - f + 1)!^2 * f!) where f = floor(n/2). - Peter Luschny, Sep 30 2018

A350499 Unsigned coefficients of free moment partition polynomials determining the free cumulants from the free moments of free probability theory. Irregular triangle with row lengths given by A000041, n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 4, 2, 10, 5, 1, 5, 5, 15, 15, 35, 14, 1, 6, 6, 3, 21, 42, 7, 56, 84, 126, 42, 1, 7, 7, 7, 28, 56, 28, 28, 84, 252, 84, 210, 420, 462, 132, 1, 8, 8, 8, 4, 36, 72, 72, 36, 36, 120, 360, 180, 360, 30, 330, 1320, 660, 792, 1980, 1716, 429

Offset: 1

Tom Copeland, Jan 01 2022

Coefficients are listed in Abramowitz and Stegun order (A036036).
Irregular triangular matrix of the unsigned coefficients of the free moment partition polynomials of free probability theory, for a single variable, that give the free formal cumulants given the free formal moments. This set of partition polynomials together with those of A134264 are the counterparts to the exp-log relations for the classical formal moments and cumulants associated with A036040 and A127671.
Associations with a compositional inverse pair of Laurent series, Kac-Schwarz operators of 2-D quantum theory, Virasoro / Witt / Heisenberg group actions, and KP and KdV integrable hierarchies are noted in references supplied in the MathOverflow link as well as a geometric combinatorial model based upon noncrossing partitions.

			Triangle begins:
  1;
  1, 1;
  1, 3, 2;
  1, 4, 2, 10,  5;
  1, 5, 5, 15, 15, 35, 14;
  ...
___________
The first few free cumulants in terms of the free moments are
  c_1 = m_1
  c_2 = m_2 - m_1^2
  c_3 = m_3 - 3 m_2 m_1 + 2 m_1^3
  c_4 = m_4 - 2 m_2^2 - 4m_3 m_1 + 10 m_2 m_1^2 - 5 m_1^4
  c_5 = m_5 - 5 m_2  m_3 - 5  m_4 m_1 + 15  m_2^2 m_1 + 15 m_3 m_1^2 - 35 m_2 m_1^3 + 14 m_1^5
___________
Conversely, from A134264, these free moments in terms of the free cumulants are
  m_1 = c_1
  m_2 = c_2 + c_1^2
  m_3 = c_3 + 3 c_2 c_1 + c_1^3
  m_4 = c_4 + + 2 c_2^2 + 4 c_3 c_1 + 6 c_2 c_1^2 + c_1^4
  m_5 = c_5 + 5 c_2 c_3 + 5 c_4 c_1 + 10 c_2^2 c_1 + 10 c_3 c_1^2  + 10 c_2 c_1^3 + c_1^5
___________

Tom Copeland, Ruling the inverse universe, the inviscid Hopf-Burgers evolution equation: Compositional inversion, free probability, associahedra, diff ids, integrable hierarchies, and translation, 2022
MathOverflow, Combinatorics for the action of Virasoro / Kac-Schwarz operators: partition polynomials of free probability theory, a MO question posed by Tom Copeland, 2021.
J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Cf. A000041, A000108, A001764, A002293, A006318, A036040, A060693, A086810, A088617, A111785, A127671, A133437, A133932, A134685, A134264, A276850.

mv(n)={eval(Str("'m",n))}
Trm(m,v)={my(S=Set(v)); for(i=1, #S, my(x=S[i]); m=polcoef(m, #select(y->y==x, v), mv(x))); m}
Q(n)={polcoef(-x/serreverse(x*(1 + sum(k=1, n, -x^k*mv(k), O(x*x^n)))), n)}
row(n)={my(q=Q(n)); [Trm(q,Vec(v)) | v<-partitions(n)]}
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022

C(v)={my(n=vecsum(v), S=Set(v)); (n+#v-2)!/(n-1)!/prod(i=1, #S, my(x=S[i]); (#select(y->y==x, v))!)}
row(n)=[C(Vec(p)) | p<-partitions(n)]
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022

O.g.f.: C(x) = 1 + c_1 x + c_2 x^2 + ... = x / (x + m_1 x^2 + m_2 x^3 + m_3 x^4 + ...)^(-1) = x / M^(-1)(x), the shifted reciprocal of the compositional inverse of a power series M(x) = x + m_1 x^2 + m_2 x^3 + ..., the o.g.f. of the free moments m_n in free probability theory.
Row sums: big Schroeder numbers A006318.
Refinement of A060693 and A088617, i.e., by letting m_n = -t and removing all resulting signs, the elements of these two lower triangular matrices are generated.
The coefficients of the highest order terms in m_1^n of the free moment partition polynomials are the signed Catalan numbers A000108.
Taking the derivative with respect to the indeterminate m_1 generates the Lagrange inversion partition polynomials, with shifted indices, of A133437 and A111785 with an overall scale factor. These Lagrange inversion polynomials are the refined Euler characteristic polynomials of the associahedra. E.g.,
  D_{m_1} c_5 = 5 (- m_4 + 3 m_2^2 + 6 m_3 m_1 - 21 m_2 m_1^2 + 14 m_1^4). An analogous differential formula that applies to the classical formal cumulants in relation to the permutahedra is stated in my 2012 comment in A127671.
The o.g.f. satisfies the partial differential equations D_{m_1} (x / C(x)) = -(1/3) D_x (x / C(x))^3 and D_{m_1} (C(x) / x) = D_x (x / C(x)), where D_{m_1} and D_x are the partial derivatives with respect to m_1 and x.
More generally, D_{m_n} (x / C(x)) = -(1/(n+2)) D_x (x / C(x))^{n+2), equivalent to  D_{m_n} M^(-1)(x) = -(1/(n+2)) D_x (M^(-1)(x))^{n+2). Equations of this type are found in Zhou (see eqn. 44 on p. 11), characterizing the KdV hierarchy. These differential equations can be transformed into the inviscid Burgers-Hopf partial differential equation (see, e.g., A133437, A086810, A001764, A002293, A133932, A134685, and A276850).
The formal free cumulants when identified as the indeterminates of the noncrossing Lagrange inversion partition polynomials NCP_n(c_1,c_2,...,c_n) = m_n of A134264 (as in the example section) satisfy the partial differential equations D_{m_k} NCP_n(c_1, ..., c_n) = d(m_n)/dm_k = delta_{n-k}, where delta_{n} is the Kronecker delta which is zero for all integers n other than n = 0, for which it evaluates as unity. This provides a recursion method for determining the partial derivatives dc_n/dm_k from the partial derivatives dc_p/dm_k and cumulants c_p with k <= p < n. For example, dc_k/dm_j = 0 for j > k and dc_k/dm_k = 1, so dm_3/dm_2 = 0 = D_{m_2} (c_3 + 3 c_2 c_1 + c_1^3) = dc_3/dm_2 + 3 c_1  dc_2/dm_2 = dc_3/dm_2 + 3 c_1 , implying dc_3/dm_2 = -3 c_1 = -3 m_1.
T(n,k) = (n+j-2)!/((n-1)!*Product_{i>=1} s_i!), where (1*s_1 + 2*s_2 + ... = n) is the k-th partition of n and j = s_1 + s_2 + ... is the number of parts. - Andrew Howroyd, Feb 01 2022
Conjecture: free cumulants in terms of the free moments are R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) - Sum_{j=1..n-1} R(j,k)*R(n-j,1) for n > 1, k > 0 with R(1,k) = m_k for k > 0. - Mikhail Kurkov, Mar 30 2025

Terms a(19) and beyond from Andrew Howroyd, Feb 01 2022

A154829 A q-Catalan triangle for q=2.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 17, 25, 9, 1, 171, 258, 102, 16, 1, 3113, 4635, 1788, 290, 25, 1, 106419, 154048, 54909, 7910, 665, 36, 1, 7035649, 9907933, 3232971, 385669, 26257, 1323, 49, 1, 915028347, 1262093470, 382948336, 37703584, 1889650, 71596, 2380, 64, 1

Offset: 0

Paul Barry, Jan 15 2009

First column is A015083. Row sums are A154828.

			Triangle begins
1,
1, 1,
3, 4, 1,
17, 25, 9, 1,
171, 258, 102, 16, 1,
3113, 4635, 1788, 290, 25, 1

Cf. A060693.

Triangle [1,2,4,8,16,32,...] DELTA [1,0,1,0,1,0,1,....] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 28 2011

G.f.: 1/(1-(x+xy)/(1-2x/(1-(4x+xy)/(1-8x/(1-(16x+xy)/(1-.... (continued fraction).

A215294 Number of permutations of 0..floor((n*3-2)/2) on odd squares of an n X 3 array such that each row and column of odd squares is increasing.

Original entry on oeis.org

1, 3, 6, 30, 70, 420, 1050, 6930, 18018, 126126, 336336, 2450448, 6651216, 49884120, 137181330, 1051723530, 2921454250, 22787343150, 63804560820, 504636071940, 1422156202740, 11377249621920, 32235540595440, 260363981732400

Offset: 1

R. H. Hardin, Aug 07 2012

nonn

a(n) is number of symmetric standard Young tableaux of shape (n,n,n). - Ran Pan, May 21 2015

			Some solutions for n=5:
  x 1 x   x 0 x   x 0 x   x 4 x   x 0 x   x 1 x   x 1 x
  0 x 5   2 x 4   2 x 5   0 x 2   1 x 2   0 x 5   0 x 3
  x 3 x   x 1 x   x 1 x   x 5 x   x 3 x   x 2 x   x 2 x
  2 x 6   3 x 6   3 x 6   1 x 3   4 x 6   3 x 6   4 x 5
  x 4 x   x 5 x   x 4 x   x 6 x   x 5 x   x 4 x   x 6 x

R. H. Hardin, Table of n, a(n) for n = 1..210
Ran Pan, Problem 4, Project P.

Column 3 of A215297.

Cf. A060693.

a := n -> `if`(irem(n, 2) = 0, ((1/2)*n+1)*factorial((3/2)*n)/ (factorial((1/2)*n+1)^2*factorial((1/2)*n)), factorial((3/2)*n+3/2)/ (factorial((1/2)*n+1/2)^3*((9/2)*n+3/2))): # Peter Luschny, Sep 30 2018

a(n) = A060854(1,f3)*A060854(2,f4)*binomial(1*f3+2*f4,1*f3) where f3 = floor((n+1)/2), f4 = floor(n/2).

a(n) = e(n) if n even otherwise o(n), where e(n) = 6*Gamma((3*n)/2)/((2 + n)*Gamma(1 + n/2)^2*Gamma(n/2)) and o(n) = (1 + n)*Gamma(1/2 + (3*n)/2)/(2*Gamma((3 + n)/2)^3). - Peter Luschny, Sep 30 2018

A319578 a(n) = (1/3)(n+2)^2(3*n+3)!/(n+2)!^3.

Original entry on oeis.org

1, 10, 140, 2310, 42042, 816816, 16628040, 350574510, 7595781050, 168212023980, 3792416540640, 86787993910800, 2011383287449200, 47123837020238400, 1114478745528638160, 26575401262863040830, 638330716607984804250, 15431925043610580004500, 375239440534109892741000

Offset: 0

Peter Luschny, Sep 30 2018

Number of Schröder paths of length 2n+1 having n peaks.

Cf. A007004, A060693, A215287.

[(1/3)*(n+2)^2*Factorial(3*n+3)/Factorial(n+2)^3: n in [0..20]]; // Vincenzo Librandi, Oct 01 2018

a := n -> (n+2)*(3*n+2)!/((n+2)!^2*n!): seq(a(n), n = 0..18);

Table[(n+2) (3*n+2)! / ((n+2)!^2 n!), {n, 0, 30}] (* Vincenzo Librandi, Oct 01 2018 *)

a(n) = (1/3)*(n+2)^2*(3*n+3)!/(n+2)!^3; \\ Michel Marcus, Oct 01 2018

a(n) = (n+2)*(3*n+2)!/((n+2)!^2*n!).
a(n) = A060693(2n+1,n).
G.f.: (hypergeom([1/3, 2/3], [2], 27*x) - 1)/(3*x). - Stefano Spezia, Aug 25 2025

cp's OEIS Frontend

A215287 Number of permutations of 0..floor((n*3-1)/2) on even squares of an n X 3 array such that each row and column of even squares is increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A350499 Unsigned coefficients of free moment partition polynomials determining the free cumulants from the free moments of free probability theory. Irregular triangle with row lengths given by A000041, n >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A154829 A q-Catalan triangle for q=2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A215294 Number of permutations of 0..floor((n*3-2)/2) on odd squares of an n X 3 array such that each row and column of odd squares is increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A319578 a(n) = (1/3)*(n+2)^2*(3*n+3)!/(n+2)!^3.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A319578 a(n) = (1/3)(n+2)^2(3*n+3)!/(n+2)!^3.