A232434 Limit of rows in triangle A232433 when read in reverse order.

1, 2, 6, 14, 32, 68, 142, 276, 542, 1022, 1876, 3394, 6066, 10628, 18412, 31344, 52868, 88370, 146180, 239310, 388370, 624688, 997586, 1582640, 2493908, 3902574, 6069194, 9378078, 14411150, 22034860, 33520642, 50747992, 76471200, 114689926, 171242092, 254587046, 376981800, 556129468, 817412048, 1197096472, 1747047580

Offset

1,2

Comments

Conjecture: a(n) equals sum of f(lambda) over all partitions of n, where f is defined recursively as f({})=1; f(lambda)=binomial(i+j,j) f(mu)f(nu); with i and j the row and column of the box in the Young-Ferrers diagram of lambda such that i+j is maximized, and mu is lambda with the first i rows removed, and nu is lambda with the first j columns removed. See Math Overflow link. - Wouter Meeussen, Apr 07 2014

Links

Table of n, a(n) for n=1..41.

Matt Fayers, A function from partitions to natural numbers - is it familiar?, MathOverflow 30 may 2013. [From Wouter Meeussen, Apr 07 2014]

Formula

E.g.f. of triangle A232433 satisfies: G(x,q) = exp(Integral G(x,q)*G(q*x,q) dx).

Example

The triangle A232433 of coefficients of x^n*q^k, n >= 0, k = 0..n*(n-1)/2, begins:
[1];
[1];
[2, 1];
[6, 6, 2, 1];
[24, 36, 22, 14, 6, 2, 1];
[120, 240, 210, 160, 104, 56, 32, 14, 6, 2, 1];
[720, 1800, 2040, 1830, 1448, 992, 674, 408, 232, 128, 68, 32, 14, 6, 2, 1]; ...
where this sequence is the limit of the rows read in reverse order.

Mathematica

Clear[c]; c[0] = 1; Table[f = Sum[c[k] x^k/k!, {k, 0, n}];
c[n + 1] = n! SeriesCoefficient[f^2 (f /. x -> q x), {x, 0, n}] // Simplify;  Coefficient[q*c[n + 1], q^(1 + n*(n - 1)/2)], {n, 0, 64}]
(* or via combinatorics: *)
Clear[f]; f[{}]:=1; f[\[Lambda]_?PartitionQ]:=f[\[Lambda]]=Block[{temp, i, j, \[Mu], \[Nu]}, temp=\[Lambda]+Range[Length[\[Lambda]]]; {i}=First@Position[temp, Max[temp], 1, 1]; j=\[Lambda][[i]]; \[Mu]=Drop[\[Lambda], i]; \[Nu]=DeleteCases[\[Lambda]-j, q_/; (q<=0)]; Binomial[i+j, j]f[\[Mu]]f[\[Nu]]];
Table[Total[f/@IntegerPartitions[n]], {n, 0, 24}] (* Wouter Meeussen, Apr 07 2014 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(intformal(A*subst(A, x, x*y +x*O(x^n)), x))); n!*polcoeff(polcoeff(A, n, x), (n-1)*(n-2)/2, y)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A232433.

Sequence in context: A301554 A217941 A346679 * A096238 A074878 A065495

Adjacent sequences: A232431 A232432 A232433 * A232435 A232436 A232437

Keyword

nonn

Author

Paul D. Hanna, Nov 23 2013

Status

approved