A300581
Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).
Original entry on oeis.org
1, 4, 24, 112, 544, 2368, 10624, 44800, 190976, 791552, 3282944, 13414400, 54829056, 222117888, 899383296, 3625123840, 14601027584, 58659700736, 235555782656, 944552017920, 3786334535680, 15166305468416, 60736264994816, 243129089261568, 973133053952000
Offset: 0
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nmax = 25; CoefficientList[Series[Product[1/(1-2^(k+1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
A134135
Alternating row sums of triangle A134134.
Original entry on oeis.org
1, 1, 5, 15, 93, 551, 4129, 33607, 312929, 3179343, 35602881, 432201743, 5678740945, 80142780751, 1210609725905, 19481112885231, 332836223507793, 6016678424942063, 114746996449871761, 2302527084416470255
Offset: 1
A158615
Expansion of Sum_{n>0} n*n!*x^n/(1-n!*x^n).
Original entry on oeis.org
1, 5, 19, 105, 601, 4445, 35281, 324897, 3266569, 36360065, 439084801, 5751188913, 80951270401, 1220673888257, 19615124183329, 334777645154817, 6046686277632001, 115243914079782593, 2311256907767808001
Offset: 1
-
nmax := 40: gf := add( taylor( n*n!*x^n/(1-n!*x^n),x=0,nmax+1),n=1..nmax ) : coeffs(convert(gf,polynom)) ; # R. J. Mathar, Mar 30 2009
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nmax=20; Rest[CoefficientList[Series[Sum[k*k!*x^k/(1-k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 19 2015 *)
A160564
Sum of products of factorials of parts times the factorial of the number of parts in all integer partitions of n.
Original entry on oeis.org
1, 1, 4, 16, 80, 420, 2592, 17352, 132240, 1117200, 10559040, 110276352, 1268640000, 15923168640, 216767367936, 3178157607936, 49918919122944, 835744605027840, 14852897362759680, 279172076525153280, 5531978038112409600, 115241366146485749760
Offset: 0
a(3) = 16 because the partitions of 3 can be so ordered in 16 ways: 3 (6); 2,1 (4); 1,1,1 (6).
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b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
add(b(n-i*j, i-1, p+j)*i!^j, j=0..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..23); # Alois P. Heinz, Oct 02 2017
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p = Table[Map[Function[n, Apply[Times, n! ]], Partitions[i]], {i, 0, 20}]; q = Table[Map[Function[n, Length[n]! ], Partitions[i]], {i, 0, 20}]; Map[Function[n, Apply[Plus, n]], p*q]
A292318
Expansion of Product_{k>=1} ((1 + k!*x^k)/(1 - k!*x^k)).
Original entry on oeis.org
1, 2, 6, 22, 90, 434, 2442, 15874, 118722, 1009586, 9640866, 102243682, 1191949122, 15141785570, 208068223458, 3073613823778, 48554040330210, 816547584905186, 14562214993474914, 274463503469613538, 5450631032885614050, 113749623991878727394
Offset: 0
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nmax = 25; CoefficientList[Series[Product[(1 + k!*x^k)/(1 - k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 15 2017 *)
A318247
a(n) = [x^n] Product_{k>=1} (1 + n!*x^k).
Original entry on oeis.org
1, 1, 2, 42, 600, 28920, 374285520, 128100273840, 131101518683520, 143354704247556480, 173401404266683545849388800, 2538767479410416957720411116800, 105287752487031026606448840363801600, 4510685217145833106538730603088118860800, 288804138719404983322786510403231912442931200
Offset: 0
-
nmax = 15; Table[SeriesCoefficient[Product[(1+n!*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
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