A351433
a(n) = n! * [x^n] 1/(1 + f^n(x)), where f(x) = exp(x) - 1.
Original entry on oeis.org
1, -1, 0, 0, -2, -75, -3334, -192864, -14443260, -1372372623, -162009663365, -23314158802286, -4022712394579207, -820399656345934444, -195326656416326556562, -53709209673236813446542, -16896296201917398543629108, -6030879950631118091070849321
Offset: 0
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g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1/(1+(g@@n)(x)), x, n+1), x, n):
seq(a(n), n=0..22); # 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, n]; Array[a, 17, 0] (* 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)));
a(n) = T(n, n);
A111933
Triangle read by rows, generated from Stirling cycle numbers.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 15, 35, 24, 1, 5, 26, 105, 228, 120, 1, 6, 40, 234, 947, 1834, 720, 1, 7, 57, 440, 2696, 10472, 17582, 5040, 1, 8, 77, 741, 6170, 37919, 137337, 195866, 40320, 1, 9, 100, 1155, 12244, 105315, 630521, 2085605, 2487832, 362880
Offset: 1
Row 5 of the triangle = 1, 4, 15, 35, 24; generated from M^n * [1,0,0,0,...] (n = 1 through 5); then take antidiagonals.
Terms in the array, first few rows are:
1, 1, 2, 6, 24, 120, ...
1, 2, 7, 35, 228, 1834, ...
1, 3, 15, 105, 947, 10472, ...
1, 4, 26, 234, 2697, 37919, ...
1, 5, 40, 440, 6170, 105315, ...
1, 6, 57, 741, 12244, 245755, ...
...
First few rows of the triangle are:
1;
1, 1;
1, 2, 2;
1, 3, 7, 6;
1, 4, 15, 35, 24;
1, 5, 26, 105, 228, 120;
1, 6, 40, 234, 947, 1834, 720;
...
A351424
a(n) = n! * [x^n] -log(1 - f^(n-1)(x)), where f(x) = log(1+x).
Original entry on oeis.org
1, 0, 3, -48, 1270, -50375, 2803829, -208616562, 20003317746, -2402323535658, 353219463307920, -62411008199372327, 13048469028962425266, -3186116313706825820802, 898478811755719496052919, -289795933163271680910773018, 106008143082108931457543700504
Offset: 1
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g:= x-> log(1+x):
a:= n-> n! * coeff(series(-log(1-(g@@(n-1))(x)), x, n+1), x, n):
seq(a(n), n=1..19); # Alois P. Heinz, Feb 11 2022
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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, n]; Array[a, 16] (* Amiram Eldar, Feb 11 2022 *)
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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, n);
Showing 1-3 of 3 results.
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