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)));
A351428
Expansion of e.g.f. 1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1).
Original entry on oeis.org
1, -1, -3, -11, -41, -75, 1540, 37725, 657715, 10551750, 163089430, 2407275470, 31865298262, 290682880132, -2479867505029, -267542605513289, -11438897571729494, -404343336811199242, -13192591498632627584, -410340915410006575406, -12233989907129223814578
Offset: 0
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g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1/((g@@5)(x)+1), x, n+1), x, n):
seq(a(n), n=0..20); # 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}]; a[n_] := T[n, 5]; Array[a, 20, 0] (* Amiram Eldar, Feb 11 2022 *)
With[{nn=20},CoefficientList[Series[1/Exp[Exp[Exp[Exp[Exp[x]-1]-1]-1]-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 09 2025 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/exp(exp(exp(exp(exp(x)-1)-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, 5);
A363008
Expansion of e.g.f. 1/(2 - exp(exp(exp(exp(x) - 1) - 1) - 1)).
Original entry on oeis.org
1, 1, 6, 52, 594, 8444, 143783, 2854261, 64735570, 1651560175, 46814933977, 1459689346911, 49650414218071, 1829560770160335, 72603137881845927, 3086932915850946633, 139999909097319319787, 6746170002325663539844, 344199636595620793896784
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, 4):
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(x)-1)-1)-1))))
A351422
Expansion of e.g.f. -log(1 - log(1 + log(1 + log(1+x)))).
Original entry on oeis.org
1, -2, 8, -48, 386, -3905, 47701, -683592, 11250291, -209168071, 4336482905, -99197868847, 2481962140797, -67426166949102, 1976463051528507, -62178381389729317, 2089532143617395264, -74702625442877063902, 2830904065389397804534, -113348477836878447492630
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, 4]; Array[a, 20] (* Amiram Eldar, Feb 11 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(-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, 4);
Showing 1-4 of 4 results.