A349557
E.g.f. satisfies: log(A(x)) = (exp(x*A(x)) - 1) * A(x).
Original entry on oeis.org
1, 1, 6, 68, 1163, 26787, 778128, 27325321, 1126308870, 53323302708, 2851990661789, 170088808988705, 11192134680722586, 805521092432042573, 62950026461699015998, 5308512876799649771192, 480492707646769163920059, 46464318322169305448661915
Offset: 0
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a[n_] := Sum[(n + k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 22 2021 *)
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a(n) = sum(k=0, n, (n+k+1)^(k-1)*stirling(n, k, 2));
A356785
E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x).
Original entry on oeis.org
1, 0, 2, 3, 64, 365, 7356, 85687, 1920752, 34821369, 905128300, 22172123171, 672107454888, 20552960420005, 721088019634724, 26257726364294895, 1053711696230404576, 44336326818388565105, 2010106841636689325532, 95747319823049127621019
Offset: 0
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nmax = 19; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x*A[x]]-1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
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a(n) = n!*sum(k=0, n\2, (n+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A356788
E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x)^2.
Original entry on oeis.org
1, 0, 2, 3, 88, 485, 13896, 158767, 4919664, 90698841, 3130084360, 81025744811, 3144372342552, 104942286748741, 4582896912897408, 186591555463556895, 9135453970592830816, 437146665470130792497, 23852990622867670807704, 1307029600226135900982835
Offset: 0
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a(n) = n!*sum(k=0, n\2, (n+k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A356789
E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x)^3.
Original entry on oeis.org
1, 0, 2, 3, 112, 605, 22596, 254527, 10166416, 188035353, 8190917380, 217293592571, 10408915205976, 363500829796117, 19203682103461324, 833182131498018135, 48525371633295259936, 2511705297938365594289, 160874324235464440678164
Offset: 0
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a(n) = n!*sum(k=0, n\2, (n+2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A356882
E.g.f. satisfies: A(x) * log(A(x)) = x * (exp(x*A(x)) - 1).
Original entry on oeis.org
1, 0, 2, 3, 16, 125, 756, 7567, 85968, 994905, 14373460, 225366251, 3800667960, 72169966453, 1469546796732, 32150706096615, 760806334538656, 19142440567996721, 512272692571487652, 14560087915617858883, 436598686303562722440, 13796641165956117509901
Offset: 0
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a(n) = n!*sum(k=0, n\2, (n-2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A356883
E.g.f. satisfies: A(x)^2 * log(A(x)) = x * (exp(x*A(x)) - 1).
Original entry on oeis.org
1, 0, 2, 3, -8, 5, 696, 2527, -40144, -178407, 8337880, 76134971, -1781542344, -24938260763, 691630553264, 14216543752335, -312910463346464, -9343318015483471, 195539694928047144, 8145971436703039363, -142317653823753257560, -8498984155838272275459
Offset: 0
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a(n) = n!*sum(k=0, n\2, (n-3*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A375827
E.g.f. satisfies A(x) = exp(x^2 * (exp(x*A(x)) - 1)).
Original entry on oeis.org
1, 0, 0, 6, 12, 20, 1110, 10122, 58856, 1239912, 23773770, 303367790, 5442263772, 135931189836, 2868815413646, 61708462976370, 1696559552295120, 48121891688969552, 1323641220801211026, 40678367069424137814, 1372011621943725532100
Offset: 0
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a(n) = n!*sum(k=0, n\3, (n-2*k+1)^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!);
A375831
E.g.f. satisfies A(x) = exp(x * (exp(x^2*A(x)) - 1)).
Original entry on oeis.org
1, 0, 0, 6, 0, 60, 1080, 840, 80640, 982800, 5292000, 249812640, 2854051200, 46711304640, 1595483809920, 22132648137600, 649972279756800, 19151306772998400, 377272414943424000, 14076577060273728000, 407012458114918656000, 11429334092933569612800
Offset: 0
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Table[n! * Sum[(k+1)^(n-2*k-1) * StirlingS2[k,n-2*k]/k!, {k,0,Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Aug 31 2024 *)
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a(n) = n!*sum(k=0, n\2, (k+1)^(n-2*k-1)*stirling(k, n-2*k, 2)/k!);
Showing 1-8 of 8 results.