A336020
a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} Stirling2(n-3,k) * a(k).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 5, 15, 53, 222, 1115, 6698, 47243, 382187, 3480048, 35251942, 394839407, 4875966656, 66282636371, 989985346269, 16198580140543, 289168351452220, 5604120791540468, 117309414122840454, 2639927837211705159, 63618153549702851338
Offset: 0
-
b:= proc(n, m) option remember; `if`(n=0,
a(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> `if`(n<3, 1, b(n-3, 0)):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 13 2021
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a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = 1 + x + x^2/2 + Integrate[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
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lista(nn) = {my(va = vector(nn, k, 1)); for (n=4, nn, va[n] = sum(k=0, n-3, stirling(n-4, k, 2)*va[k+1]);); va;} \\ Michel Marcus, Jul 06 2020
A336021
a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} Stirling2(n-4,k) * a(k).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 2, 5, 15, 52, 204, 902, 4532, 26196, 175320, 1351296, 11819348, 115309534, 1236465988, 14419850138, 181652022376, 2462053028798, 35834756184146, 559816444117400, 9389648056139010, 169166236946379128, 3273760080403458226, 67994123544008546820
Offset: 0
-
b:= proc(n, m) option remember; `if`(n=0,
a(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> `if`(n<4, 1, b(n-4, 0)):
seq(a(n), n=0..27); # Alois P. Heinz, Aug 13 2021
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a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 4, k] a[k], {k, 0, n - 4}]; Table[a[n], {n, 0, 27}]
nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2/2 + x^3/6 + Integrate[Integrate[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
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lista(nn) = {my(va = vector(nn, k, 1)); for (n=5, nn, va[n] = sum(k=0, n-4, stirling(n-5, k, 2)*va[k+1]);); va;} \\ Michel Marcus, Jul 06 2020
A336022
a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} Stirling2(n-5,k) * a(k).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, 5, 15, 52, 203, 878, 4172, 21767, 125536, 809254, 5890115, 48560551, 450859572, 4657423009, 52802518648, 649162712358, 8574743501046, 120876064485660, 1809924607067234, 28694297293078915, 480719498205658859, 8502406681853097237
Offset: 0
-
b:= proc(n, m) option remember; `if`(n=0,
a(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> `if`(n<5, 1, b(n-5, 0)):
seq(a(n), n=0..28); # Alois P. Heinz, Aug 13 2021
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a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 5, k] a[k], {k, 0, n - 5}]; Table[a[n], {n, 0, 28}]
nmax = 28; A[] = 0; Do[A[x] = 1 + x + x^2/2 + x^3/6 + x^4/24 + Integrate[Integrate[Integrate[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
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lista(nn) = {my(va = vector(nn, k, 1)); for (n=6, nn, va[n] = sum(k=0, n-5, stirling(n-6, k, 2)*va[k+1]);); va;} \\ Michel Marcus, Jul 05 2020
A345177
a(0) = 1, a(1) = 0; a(n+2) = Sum_{k=0..n} Stirling2(n,k) * a(k).
Original entry on oeis.org
1, 0, 1, 0, 1, 3, 8, 28, 149, 1029, 8039, 69375, 675541, 7584630, 98484836, 1457695370, 24117255106, 439505090491, 8756668806615, 190293641816660, 4508138040317573, 116298682305458460, 3258081214212853975, 98709283556190931672, 3219222306795403565116, 112538217720491999726102
Offset: 0
-
b:= proc(n, m) option remember; `if`(n=0,
a(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> `if`(n<2, 1-n, b(n-2, 0)):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 13 2021
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a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[StirlingS2[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = 1 + Normal[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!
A345178
a(0) = 0, a(1) = 1; a(n+2) = Sum_{k=0..n} Stirling2(n,k) * a(k).
Original entry on oeis.org
0, 1, 0, 1, 1, 2, 8, 38, 194, 1138, 8154, 71544, 739406, 8674238, 113451160, 1648133190, 26631054962, 478633871152, 9531297220728, 208851860234540, 4997665703050398, 129765874491438094, 3639593254921626678, 109942671192206473592, 3569449102675488493032, 124319448405579907085938
Offset: 0
-
b:= proc(n, m) option remember; `if`(n=0,
a(m), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> `if`(n<2, n, b(n-2, 0)):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 13 2021
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a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 2, k] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = x + Normal[Integrate[Integrate[A[Exp[x] - 1 + O[x]^(nmax + 1)], x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!
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