A379994
Expansion of e.g.f. exp(-3*x)/(exp(-x) - x)^4.
Original entry on oeis.org
1, 5, 37, 349, 3969, 52641, 796069, 13502693, 253667297, 5225690017, 117090480021, 2834363683317, 73697918467105, 2048252470006913, 60587779740857573, 1900347489736371301, 62992469337321611073, 2200238756416513719489, 80765631192992760237205
Offset: 0
A379995
Expansion of e.g.f. exp(-2*x)/(exp(-x) - x)^4.
Original entry on oeis.org
1, 6, 48, 476, 5608, 76372, 1179016, 20332580, 387225120, 8068825988, 182564048824, 4456476380644, 116724944900272, 3264981100202564, 97130013288324552, 3062011655207131748, 101963095705628194624, 3576126056313566090500, 131762871920106615643480
Offset: 0
A379996
Expansion of e.g.f. exp(-x)/(exp(-x) - x)^4.
Original entry on oeis.org
1, 7, 61, 639, 7825, 109683, 1731645, 30403495, 587595649, 12395233539, 283385424829, 6979650164391, 184235963026833, 5188528445210035, 155284012799863453, 4921569063327156807, 164672737994453759617, 5800532536265417597571, 214559605389429001486557
Offset: 0
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With[{nn=20},CoefficientList[Series[Exp[-x]/(Exp[-x]-x)^4,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2025 *)
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a(n) = n!*sum(k=0, n, (k+3)^(n-k)*binomial(k+3, 3)/(n-k)!);
A380841
Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 21, 10, 3, 1, 0, 148, 66, 18, 4, 1, 0, 1305, 560, 141, 28, 5, 1, 0, 13806, 5770, 1380, 252, 40, 6, 1, 0, 170401, 69852, 16095, 2776, 405, 54, 7, 1, 0, 2403640, 970886, 217458, 35940, 4940, 606, 70, 8, 1, 0, 38143377, 15228880, 3335745, 533304, 70045, 8088, 861, 88, 9, 1
Offset: 0
Array begins as:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 4, 10, 18, 28, 40, 54, ...
0, 21, 66, 141, 252, 405, 606, ...
0, 148, 560, 1380, 2776, 4940, 8088, ...
0, 1305, 5770, 16095, 35940, 70045, 124350, ...
...
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A[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten
Showing 1-4 of 4 results.