A351429
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + f^k(x)), where f(x) = exp(x) - 1.
Original entry on oeis.org
1, 1, -1, 1, -1, 2, 1, -1, 1, -6, 1, -1, 0, -1, 24, 1, -1, -1, 1, 1, -120, 1, -1, -2, 0, 1, -1, 720, 1, -1, -3, -4, 6, -2, 1, -5040, 1, -1, -4, -11, -2, 32, -9, -1, 40320, 1, -1, -5, -21, -41, 76, 115, -9, 1, -362880, 1, -1, -6, -34, -129, -75, 953, 172, 50, -1, 3628800
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
-1, -1, -1, -1, -1, -1, -1, ...
2, 1, 0, -1, -2, -3, -4, ...
-6, -1, 1, 0, -4, -11, -21, ...
24, 1, 1, 6, -2, -41, -129, ...
-120, -1, -2, 32, 76, -75, -806, ...
720, 1, -9, 115, 953, 1540, -3334, ...
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A:= (n, k)-> n!*(g->coeff(series(1/(1+(g@@k)(x)), x, n+1), x, n))(x->exp(x)-1):
seq(seq(A(n, d-n), n=0..d), d=0..10); # Alois P. Heinz, Feb 11 2022
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T[n_, 0] := (-1)^n*n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 11 2022 *)
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T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
A351427
Expansion of e.g.f. 1/exp(exp(exp(exp(x)-1)-1)-1).
Original entry on oeis.org
1, -1, -2, -4, -2, 76, 953, 9103, 77054, 550457, 2123247, -32551171, -1197444063, -26019611323, -478608682879, -7915791047153, -115777452314939, -1320533985179144, -3550854626237496, 455708391448493954, 21276221692251262984, 703173682906460544467
Offset: 0
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T[n_, 0] := (-1)^n * n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; a[n_] := T[n, 4]; Array[a, 22, 0] (* Amiram Eldar, Feb 11 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/exp(exp(exp(exp(x)-1)-1)-1)))
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T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
a(n) = T(n, 4);
A363009
Expansion of e.g.f. 1/(2 - exp(exp(exp(exp(exp(x) - 1) - 1) - 1) - 1)).
Original entry on oeis.org
1, 1, 7, 71, 949, 15775, 313920, 7279795, 192828745, 5744627550, 190131836270, 6921735519110, 274885665920198, 11826225289547024, 547926995688877245, 27199542114163170649, 1440220170795372833970, 81026116511855753816058
Offset: 0
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b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=1, m!,
b(m, 0, t-1)), m*b(n-1, m, t)+b(n-1, m+1, t))
end:
a:= n-> b(n, 0, 5):
seq(a(n), n=0..20); # Alois P. Heinz, May 12 2023
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(exp(exp(exp(exp(x)-1)-1)-1)-1))))
A351423
Expansion of e.g.f. -log(1 - log(1 + log(1 + log(1 + log(1+x))))).
Original entry on oeis.org
1, -3, 16, -124, 1270, -16243, 249776, -4494334, 92716855, -2158505443, 55996266046, -1602132913687, 50124833578256, -1702501170925098, 62391472267252322, -2453892459756494459, 103101294099324376489, -4608723131704380915202
Offset: 1
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T[n_, 1] := (n - 1)!; T[n_, k_] := T[n, k] = Sum[StirlingS1[n, j] * T[j, k - 1], {j, 1, n}]; a[n_] := T[n, 5]; Array[a, 18] (* Amiram Eldar, Feb 11 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1-log(1+log(1+log(1+log(1+x)))))))
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T(n, k) = if(k==1, (n-1)!, sum(j=1, n, stirling(n, j, 1)*T(j, k-1)));
a(n) = T(n, 5);
Showing 1-4 of 4 results.