A263576
Stirling transform of Fibonacci numbers (A000045).
Original entry on oeis.org
0, 1, 2, 6, 23, 101, 490, 2597, 14926, 92335, 610503, 4288517, 31848677, 249044068, 2043448968, 17540957166, 157108128963, 1464813176354, 14187155168782, 142469605397465, 1480903718595721, 15908940627242898, 176382950500197589, 2015650339677868116
Offset: 0
-
b:= proc(n, m) option remember; `if`(n=0, (<<0|1>,
<1|1>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..23); # Alois P. Heinz, Aug 03 2021
-
Table[Sum[Fibonacci[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
Table[Simplify[(BellB[n, GoldenRatio] - BellB[n, 1 - GoldenRatio])/Sqrt[5]], {n, 0, 23}]
A323631
Stirling transform of Pell numbers (A000129).
Original entry on oeis.org
0, 1, 3, 12, 57, 305, 1798, 11531, 79707, 589426, 4634471, 38547861, 337734048, 3105588629, 29877483743, 299906019892, 3133423928557, 34002824654365, 382507638525838, 4452923233600903, 53561431659306039, 664728428775177890, 8500763141347126563, 111886109022440334593, 1513989730079050155936
Offset: 0
-
b:= proc(n, m) option remember; `if`(n=0,
(<<2|1>, <1|0>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..24); # Alois P. Heinz, Jun 23 2023
-
FullSimplify[nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] Sinh[Sqrt[2] (Exp[x] - 1)]/Sqrt[2], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[StirlingS2[n, k] Fibonacci[k, 2], {k, 0, n}], {n, 0, 24}]
Table[Sum[Binomial[n, k] BellB[n - k] (BellB[k, Sqrt[2]] - BellB[k, -Sqrt[2]])/(2 Sqrt[2]), {k, 0, n}], {n, 0, 24}]
A323632
Stirling transform of Jacobsthal numbers (A001045).
Original entry on oeis.org
0, 1, 2, 7, 31, 152, 813, 4741, 29956, 203305, 1470795, 11276718, 91221419, 775677177, 6910797962, 64326920851, 623981351195, 6293426736344, 65867162316433, 714062197266081, 8005397253530924, 92676194887133693, 1106385117766336919, 13603803900252612966, 172082332173918135687
Offset: 0
-
b:= proc(n, m) option remember;
`if`(n=0, round(2^m/3), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..24); # Alois P. Heinz, Aug 06 2021
-
nmax = 24; CoefficientList[Series[(Exp[2 (Exp[x] - 1)] - Exp[1 - Exp[x]])/3, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] (2^k - (-1)^k)/3, {k, 0, n}], {n, 0, 24}]
Table[(BellB[n, 2] - BellB[n, -1])/3, {n, 0, 24}]
A307361
Expansion of e.g.f. (sinh(x) + 5*cosh(x) - 5)/(3 - 2*cosh(x)).
Original entry on oeis.org
0, 1, 5, 7, 65, 151, 2105, 6847, 127265, 532231, 12365705, 63206287, 1762220465, 10645162711, 346257393305, 2413453999327, 89717615769665, 708721089607591, 29639206807284905, 261679010699505967, 12159552732032614865, 118654880542567722871, 6064946899313607640505
Offset: 0
-
a:=series((sinh(x)+5*cosh(x)-5)/(3-2*cosh(x)),x=0,23):seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Apr 12 2019
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nmax = 22; CoefficientList[Series[(Sinh[x] + 5 Cosh[x] - 5)/(3 - 2 Cosh[x]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Sum[j! LucasL[j] x^j/Product[(1 + k x), {k, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) StirlingS2[n, k] k! LucasL[k], {k, 1, n}], {n, 0, 22}]
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my(x = 'x + O('x^30)); concat(0, Vec(serlaplace((sinh(x) + 5*cosh(x) - 5)/(3 - 2*cosh(x))))) \\ Michel Marcus, Apr 05 2019
Showing 1-4 of 4 results.