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A336345 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2 * (exp(x) - Sum_{j=0..k} x^j/j!)). 1
1, 1, 2, 1, 0, 6, 1, 0, 2, 22, 1, 0, 0, 2, 94, 1, 0, 0, 2, 14, 454, 1, 0, 0, 0, 2, 42, 2430, 1, 0, 0, 0, 2, 2, 222, 14214, 1, 0, 0, 0, 0, 2, 42, 1066, 89918, 1, 0, 0, 0, 0, 2, 2, 142, 6078, 610182, 1, 0, 0, 0, 0, 0, 2, 2, 366, 36490, 4412798, 1, 0, 0, 0, 0, 0, 2, 2, 142, 3082, 238046, 33827974 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
E.g.f. of column k: (Product_{j>k} exp(x^j/j!))^2.
T(0,k) = 1, T(1,k) = T(2,k) = ... = T(k,k) = 0 and T(n,k) = 2 * Sum_{j=k..n-1} binomial(n-1,j)*T(n-1-j,k) for n > k.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 0, 0, 0, 0, 0, 0, ...
6, 2, 0, 0, 0, 0, 0, ...
22, 2, 2, 0, 0, 0, 0, ...
94, 14, 2, 2, 0, 0, 0, ...
454, 42, 2, 2, 2, 0, 0, ...
2430, 222, 42, 2, 2, 2, 0, ...
PROG
(PARI) {T(n, k) = n!*polcoef(prod(j=k+1, n, exp((x^j+x*O(x^n))/j!))^2, n)}
(Ruby)
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
ary = [1]
(1..n).each{|i| ary << 2 * (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
ary
end
def A336345(n)
a = []
(0..n).each{|i| a << A(i, n - i)}
ary = []
(0..n).each{|i|
(0..i).each{|j|
ary << a[i - j][j]
}
}
ary
end
p A336345(20)
CROSSREFS
Columns k=0..4 give A001861, A194689, A339014, A339017, A339027.
Main diagonal gives A000007.
Cf. A293024.
Sequence in context: A137477 A181297 A196776 * A157982 A119275 A129462
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Nov 20 2020
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)