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A232434 Limit of rows in triangle A232433 when read in reverse order. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A327050 A301554 A217941 * A096238 A074878 A065495

Adjacent sequences:  A232431 A232432 A232433 * A232435 A232436 A232437

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 23 2013

STATUS

approved

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Last modified April 6 04:52 EDT 2020. Contains 333267 sequences. (Running on oeis4.)