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Number of plane partitions of 2n into parts of exactly n sorts which are introduced in ascending order.
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3
1, 3, 48, 1253, 45040, 2074266, 115308621, 7403931515, 542578637369, 44353623326199, 3992458392860603, 392255543503496555, 41726405940340028501, 4768006168373548992878, 582709500037368041005243, 75765509130126834789261446, 10436240655486571146294062847
Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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17
1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 7, 1, 0, 7, 30, 33, 11, 1, 0, 11, 72, 130, 77, 16, 1, 0, 15, 160, 463, 438, 157, 22, 1, 0, 22, 351, 1557, 2216, 1223, 289, 29, 1, 0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1, 0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1
COMMENTS
In general, column k>1 is asymptotic to c*k^n, where c = 1/(k!*Product_{n>=1} (1-1/k^n)) = 1/(k!*QPochhammer[1/k, 1/k]). - Vaclav Kotesovec, Jun 01 2015
EXAMPLE
T(3,1) = 3: 1a1a1a, 2a1a, 3a.
T(3,2) = 4: 1a1a1b, 1a1b1a, 1a1b1b, 2a1b.
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 3, 4, 1;
0, 5, 12, 7, 1;
0, 7, 30, 33, 11, 1;
0, 11, 72, 130, 77, 16, 1;
0, 15, 160, 463, 438, 157, 22, 1;
0, 22, 351, 1557, 2216, 1223, 289, 29, 1;
0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1;
0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000007, A000041 (for n>0), A258457, A258458, A258459, A258460, A258461, A258462, A258463, A258464, A258465.
Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.
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38
4, 0, 6, 3, 7, 5, 7, 3, 9, 9, 5, 9, 9, 5, 9, 9, 0, 7, 6, 7, 6, 9, 5, 8, 1, 2, 4, 1, 2, 4, 8, 3, 9, 7, 5, 8, 2, 1, 0, 9, 9, 7, 5, 7, 5, 1, 8, 1, 1, 4, 0, 6, 3, 5, 0, 0, 0, 4, 9, 5, 4, 8, 8, 3, 0, 3, 9, 1, 5, 0, 1, 5, 1, 8, 3, 8, 1, 2, 0, 4, 9, 7, 6, 7, 2, 5, 0, 0, 7, 2, 3, 3, 8, 1, 5, 5, 9, 2, 8, 5, 8, 2, 9, 3, 8
FORMULA
Equals LambertW(-2*exp(-2)).
EXAMPLE
-0.4063757399599599076769581241248397582109975751811406350004954883....
MATHEMATICA
RealDigits[N[ProductLog[-2/E^2], 105]][[1]] (* corrected by Vaclav Kotesovec, Feb 21 2014 *)
PROG
(PARI) solve(x=-1, x=0, x*exp(x) + 2*exp(-2)) \\ G. C. Greubel, Nov 15 2017
CROSSREFS
Cf. A106533, A187655, A187657, A217899, A217900, A226750, A243227, A245109, A247238, A258467, A337458.
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