A085686
Inverse Euler transform of Bell numbers.
Original entry on oeis.org
1, 1, 3, 9, 34, 135, 610, 2965, 15612, 87871, 526274, 3334850, 22270254, 156172689, 1146640394, 8791424549, 70227355786, 583283741066, 5027823752930, 44903579626132, 414877600876638, 3959945232723603, 38996757506464858, 395749369598406027, 4134132167178705732
Offset: 1
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read transforms; A := series(exp(exp(x)-1),x,60); A000110 := n->n!*coeff(A,x,n); [seq(A000110(i),i=1..30)]; EULERi(%);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(combinat:-bell):
seq(a(n), n = 1..25); # Peter Luschny, Nov 21 2022
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n=24; eq[0] = Rest[ Thread[ CoefficientList[ 1 + Series[ Sum[ BellB[k]*x^k, {k, 1, n}] - Product[1/(1-x^k)^a[k], {k, 1, n}], {x, 0, n}], x] == 0]]; s[1] = First[ Solve[ First[eq[0]], a[1]]]; Do[eq[k] = Rest[eq[k-1]] /. s[k]; s[k+1] = First[ Solve[ First[eq[k]], a[k+1]]], {k, 1, n-1}]; Table[a[k], {k, 1, n}] /. Flatten[Table[s[k], {k, 1, n}]] (* Jean-François Alcover, Jul 26 2011 *)
bb = Array[BellB, n = 25]; s = {}; For[i = 1, i <= n, i++, AppendTo[s, i* bb[[i]] - Sum[s[[d]]*bb[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, Apr 15 2016 *)
A305850
Weigh transform of the Bell numbers (A000110).
Original entry on oeis.org
1, 1, 2, 7, 21, 78, 305, 1304, 6007, 29854, 159012, 904986, 5479078, 35150263, 238033523, 1695554145, 12663533586, 98881246850, 805128085616, 6820302066048, 59983405937707, 546690232627480, 5154757226832625, 50208266917662433, 504482106565647708
Offset: 0
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g:= proc(n) option remember; `if`(n=0, 1,
add(binomial(n-1, j-1)*g(n-j), j=1..n))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
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g[n_] := g[n] = If[n == 0, 1,
Sum[Binomial[n - 1, j - 1]*g[n - j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
Sum[Binomial[g[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)
A305853
Inverse Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).
Original entry on oeis.org
1, 3, 10, 62, 446, 3975, 41098, 484152, 6390488, 93419965, 1498268466, 26159940522, 494036061550, 10035451747919, 218207845446062, 5057251219752612, 124462048466812950, 3241773988594489244, 89093816361187396674, 2576652694087236419386, 78224564280680539732266
Offset: 1
-
g:= proc(n) option remember; `if`(n=0, 1,
add(g(n-j)*binomial(n, j), j=1..n))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= proc(n) option remember; g(n)-b(n, n-1) end:
seq(a(n), n=1..30);
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g[n_] := g[n] = If[n == 0, 1,
Sum[g[n - j] Binomial[n, j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = g[n] - b[n, n - 1];
a /@ Range[1, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)
A353831
Product_{n>=1} (1 + a(n)*x^n) = Sum_{n>=0} Bell(n)*x^n, where Bell = A000110.
Original entry on oeis.org
1, 2, 3, 12, 34, 139, 610, 3046, 15604, 88460, 526274, 3344037, 22270254, 156359026, 1146627256, 8796070308, 70227355786, 583404596184, 5027823752930, 44907492540298, 414877525216196, 3960083715148092, 38996757506464858, 395754951565246801, 4134132167169618654, 44409616948511664062
Offset: 1
-
A[m_, n_] := A[m, n] = Which[m == 1, BellB[n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 26]
A353944
Product_{n>=1} 1 / (1 - a(n)*x^n) = Sum_{n>=0} Bell(n)*x^n, where Bell = A000110.
Original entry on oeis.org
1, 1, 3, 9, 34, 132, 610, 2929, 15604, 87310, 526274, 3325946, 22270254, 155986944, 1146627256, 8787134873, 70227355786, 583161239732, 5027823752930, 44899767806134, 414877525216196, 3959806750825202, 38996757506464858, 395743830189684984, 4134132167169618654, 44409120984298440176
Offset: 1
-
A[m_, n_] := A[m, n] = Which[m == 1, BellB[n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 26]
Showing 1-5 of 5 results.