A005494 -id:A005494 - cp's OEIS frontend

A232473 3-Fubini numbers.

6, 42, 342, 3210, 34326, 413322, 5544342, 82077450, 1330064406, 23428165002, 445828910742, 9116951060490, 199412878763286, 4646087794988682, 114884369365147542, 3005053671533400330, 82905724863616146966, 2406054103612912660362, 73277364784409578094742, 2336825320400166931304970

Offset: 3

N. J. A. Sloane, Nov 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 3..200
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.

Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A226738, A232472, A232473, A232474.

r:=3; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

# r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(3);

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 3], {n, 3, 22}] (* Jean-François Alcover, Mar 30 2016 *)

From Peter Bala, Dec 16 2020: (Start)
a(n+3) = Sum_{k = 0..n} (k+3)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+3)^n ).
a(n+3) = Sum_{k = 0..n} 3^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+3)! ).
E.g.f. with offset 0: 6*exp(3*z)/(2 - exp(z))^4 = 6 + 42*z + 342*z^2/2! + 3210*z^3/3! + .... (End)
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Dec 17 2020

A232474 4-Fubini numbers.

Original entry on oeis.org

24, 216, 2184, 24696, 310344, 4304376, 65444424, 1083832056, 19437971784, 375544415736, 7779464328264, 172062025581816, 4047849158698824, 100946105980181496, 2660400563437957704, 73890563849015945976, 2157336929022064219464, 66059202473570840113656, 2116993226046938197020744

Offset: 4

N. J. A. Sloane, Nov 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 4..200
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.

Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A232472, A232473, A232474.

r:=4; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

# r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(4);

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 4], {n, 4, 22}] (* Jean-François Alcover, Mar 30 2016 *)

From Peter Bala, Dec 16 2020: (Start)
a(n+4) = Sum_{k = 0..n} (k+4)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+4)^n ).
a(n+4) = Sum_{k = 0..n} 4^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+4)! ).
E.g.f. with offset 0: 24*exp(4*z)/(2 - exp(z))^5 = 24 + 216*z + 2184*z^2/2! + 24696*z^3/3! + .... (End)
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Dec 17 2020

A343673 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

Original entry on oeis.org

1, 4, 33, 409, 6759, 139621, 3460989, 100091335, 3308146179, 123005753041, 5081871122073, 230948185830187, 11449697796242319, 614944043618257237, 35568197580789653685, 2204201734650777596863, 145703352769994600516187, 10233323176300508748808921, 761004837938469796089586257

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000670, A005494, A006155, A032032, A343672, A343674.

a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 3 x - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (2 - 3*x - exp(x)).

a(n) ~ n! * 3^n / ((1 + LambertW(exp(2/3)/3)) * (2 - 3*LambertW(exp(2/3)/3))^(n+1)). - Vaclav Kotesovec, Jun 20 2022

A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 15, 5, 9, 13, 20, 17, 27, 37, 52, 6, 11, 16, 25, 21, 34, 47, 67, 26, 43, 60, 87, 77, 114, 151, 203, 7, 13, 19, 30, 25, 41, 57, 82, 31, 52, 73, 107, 94, 141, 188, 255, 37, 63, 89, 132, 115, 175, 235, 322, 141, 218, 295, 409, 372, 523, 674

Offset: 0

Mikhail Kurkov, Aug 23 2021 [verification needed]

Modulo 2 binomial transform of A243499(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Kevin Ryde, PARI/GP Code

Cf. A000110, A000120, A002720, A005493, A005494, A007814, A008277, A035009, A045379, A053645, A129760, A143494, A143495, A196834, A265649, A295989.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

function a = A347204(max_n)
    a(1) = 1;
    a(2) = 2;
    for nloop = 3:max_n
        n = nloop-1;
        s = 0;
        for k = 0:floor(log2(n))-1
            s = s + a(1+A053645(n)-2^k*(mod(floor(n/(2^k)),2)));
        end
        a(nloop) = 2*a(A053645(n)+1) + s;
    end
end
function a_n = A053645(n)
    a_n = n - 2^floor(log2(n));
end % Thomas Scheuerle, Oct 25 2021

f[n_] := BitAnd[n, n - 1]; a[0] = 1; a[n_] := a[n] = a[f[n]/2] + a[Floor[(n + f[n])/2]]; Array[a, 100, 0] (* Amiram Eldar, Nov 19 2021 *)

f(n) = bitand(n, n-1); \\ A129760
a(n) = if (n<=1, n+1, if (n%2, a(n\2)+a(n-1), a(f(n/2)) + a(n/2+f(n/2)))); \\ Michel Marcus, Oct 25 2021

PARI
```
\\ Also see links.
```

A129760(n) = bitand(n, n-1);
memoA347204 = Map();
A347204(n) = if (n<=1, n+1, my(v); if(mapisdefined(memoA347204,n,&v), v, v = if(n%2, A347204(n\2)+A347204(n-1), A347204(A129760(n/2)) + A347204(n/2+A129760(n/2))); mapput(memoA347204,n,v); (v))); \\ (Memoized version of Michel Marcus's program given above) - Antti Karttunen, Nov 20 2021

a(n) = a(n - 2^f(n)) + (1 + f(n))*a((n - 2^f(n))/2) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(2n+1) = a(n) + a(2n) for n >= 0.
a(2n) = a(n - 2^f(n)) + a(2n - 2^f(n)) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(n) = 2*a(f(n)) + Sum_{k=0..floor(log_2(n))-1} a(f(n) - 2^k*T(n,k)) for n > 1 with a(0) = 1, a(1) = 2, and where f(n) = A053645(n), T(n,k) = floor(n/2^k) mod 2.
Sum_{k=0..2^n - 1} a(k) = A035009(n+1) for n >= 0.
a((4^n - 1)/3) = A002720(n) for n >= 0.
a(2^n - 1) = A000110(n+1),
a(2*(2^n - 1)) = A005493(n),
a(2^2*(2^n - 1)) = A005494(n),
a(2^3*(2^n - 1)) = A045379(n),
a(2^4*(2^n - 1)) = A196834(n),
a(2^m*(2^n-1)) = T(n,m+1) is the n-th (m+1)-Bell number for n >= 0, m >= 0 where T(n,m) = m*T(n-1,m) + Sum_{k=0..n-1} binomial(n-1,k)*T(k,m) with T(0,m) = 1.
a(n) = Sum_{j=0..2^A000120(n)-1} A243499(A295989(n,j)) for n >= 0. Also A243499(n) = Sum_{j=0..2^f(n)-1} (-1)^(f(n)-f(j)) a(A295989(n,j)) for n >= 0 where f(n) = A000120(n). In other words, a(n) = Sum_{j=0..n} (binomial(n,j) mod 2)*A243499(j) and A243499(n) = Sum_{j=0..n} (-1)^(f(n)-f(j))*(binomial(n,j) mod 2)*a(j) for n >= 0 where f(n) = A000120(n).
Generalization:
b(n, x) = (1/x)*b((n - 2^f(n))/2, x) + (-1)^n*b(floor((2n - 2^f(n))/2), x) for n > 0 with b(0, x) = 1 where f(n) = A007814(n).
Sum_{k=0..2^n - 1} b(k, x) = (1/x)^n for n >= 0.
b((4^n - 1)/3, x) = (1/x)^n*n!*L_{n}(x) for n >= 0 where L_{n}(x) is the n-th Laguerre polynomial.
b((8^n - 1)/7, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A265649(n, k) for n >= 0.
b(2^n - 1, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A008277(n+1, k+1),
b(2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143494(n+2, k+2),
b(2^2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143495(n+3, k+3),
b(2^m*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*T(n+m+1, k+m+1, m+1) for n >= 0, m >= 0 where T(n,k,m) is m-Stirling numbers of the second kind.

A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2k + 3)^n / (2^k k!).

Original entry on oeis.org

1, 4, 18, 92, 532, 3440, 24552, 191280, 1612304, 14597952, 141123872, 1449324992, 15743376704, 180203389696, 2166381979264, 27274611880704, 358690234163456, 4916123783848960, 70076765972288000, 1036967662211324928, 15902394743591408640

Offset: 0

Ilya Gutkovskiy, Aug 11 2020

nonn

Cf. A004211, A005494, A007405, A337011.

nmax = 20; CoefficientList[Series[Exp[3 x + (Exp[2 x] - 1)/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k - 1) a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 20}]

E.g.f.: exp(3*x + (exp(2*x) - 1) / 2).
a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=2..n} binomial(n-1,k-1) * 2^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * A004211(k+1).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * A004211(k).

A355163 a(n) = exp(-1) * Sum_{k>=0} (4*k + 3)^n / k!.

Original entry on oeis.org

1, 7, 65, 743, 9921, 150151, 2526593, 46615783, 933072513, 20093861895, 462440842177, 11310514854375, 292627518129985, 7976748158144647, 228308400790500097, 6840702405678586343, 214000748166439723265, 6973447420429351808007, 236204029044752265931585, 8300724166287243795922151

Offset: 0

Ilya Gutkovskiy, Jun 22 2022

nonn

Cf. A000110, A005494, A126390, A284859, A284864, A285064, A355162.

nmax = 19; CoefficientList[Series[Exp[Exp[4 x] + 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] 3^(n - k) 4^k BellB[k], {k, 0, n}], {n, 0, 19}]

E.g.f.: exp(exp(4*x) + 3 x - 1).
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 4^k * Bell(k).
a(n) ~ Bell(n) * (4 + 3*LambertW(n)/n)^n. - Vaclav Kotesovec, Jun 22 2022
a(n) ~ 4^n * n^(n + 3/4) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 3/4)). - Vaclav Kotesovec, Jun 27 2022

A095674 Triangle read by rows, formed from product of Pascal's triangle (A007318) and Aitken's (or Bell's) triangle (A011971).

Original entry on oeis.org

1, 2, 2, 5, 7, 5, 15, 22, 25, 15, 52, 74, 97, 97, 52, 203, 277, 372, 449, 411, 203, 877, 1154, 1524, 1948, 2209, 1892, 877, 4140, 5294, 6816, 8734, 10718, 11570, 9402, 4140, 21147, 26441, 33255, 41954, 52357, 62107, 64404, 50127, 21147, 115975, 142416

Offset: 0

N. J. A. Sloane, based on a suggestion from Gary W. Adamson, Jun 22 2004

These triangles are to be thought of as infinite lower-triangular matrices.

			Triangle begins:
1
2 2
5 7 5
15 22 25 15
52 74 97 97 52
203 277 372 449 411 203

Cf. A007318, A011971, A095675. Row sums give A005494. First column is A000110.

a[0, 0] = 1; a[n_, 0] := a[n - 1, n - 1]; a[n_, k_] := a[n, k] = If[k < n + 1, a[n, k - 1] + a[n - 1, k - 1], 0]; p[n_, r_] := If[r <= n + 1, Binomial[n, r], 0]; am = Table[ a[n, r], {n, 0, 9}, {r, 0, 9}]; pm = Table[p[n, r], {n, 0, 9}, {r, 0, 9}]; t = Flatten[pm.am]; Delete[ t, Position[t, 0]] (* Robert G. Wilson v, Jul 12 2004 *)

More terms from Robert G. Wilson v, Jul 13 2004

A282179 E.g.f.: exp(exp(x) - 1)(exp(3x) - 2*exp(x) + 1).

Original entry on oeis.org

0, 1, 9, 52, 283, 1561, 8930, 53411, 334785, 2199034, 15119621, 108644581, 814474176, 6358910949, 51615342685, 434865155292, 3796991928727, 34308796490005, 320379418256794, 3087939032182127, 30683582797977749, 313977721545709002, 3305220440084030809, 35759627532783842561

Offset: 0

Ilya Gutkovskiy, Feb 08 2017

nonn

Stirling transform of the cubes (A000578).

Exponential convolution of the sequences A000110 and A058481 (with a(0) = 0).

			E.g.f.: A(x) = x/1! + 9*x^2/2! + 52*x^3/3! + 283*x^4/4! + 1561*x^5/5! + 8930*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..572
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's MathWorld, Stirling Transform

Cf. A000110, A000578, A005494, A033452, A058481, A186021.

b:= proc(n, m) option remember; `if`(n=0,
       m^3, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 15 2022

Range[0, 23]! CoefficientList[Series[Exp[Exp[x] - 1] (Exp[3 x] - 2 Exp[x] + 1), {x, 0, 23}], x]
Table[Sum[StirlingS2[n, k] k^3, {k, 0, n}], {n, 0, 23}]
Table[Sum[Binomial[n, k] BellB[n-k] (3^k - 2), {k, 1, n}], {n, 0, 23}]
Table[BellB[n+3] - 3*BellB[n+2] + BellB[n], {n, 0, 23}] (* Vaclav Kotesovec, Aug 06 2021 *)

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000578(k).

a(n) = A000110(n) + A005494(n) - A186021(n+1).

A338282 a(n) = (1/e^n) * Sum_{j>=3} j^n * n^j / (j-3)!.

Original entry on oeis.org

0, 4, 216, 7371, 239424, 8127875, 296315496, 11685617608, 498593804800, 22959117809685, 1137033860419000, 60338078785131785, 3418430599382500800, 206053517402599981504, 13172124530670958537160, 890361160360138336174875, 63463906792476058870550528, 4758276450884470061869230823

Offset: 0

Pedro Caceres, Oct 20 2020

nonn

			a(3) = 7371 = (1/e^3) * Sum_{j>=3} j^3 * 3^j / factorial(j-3).

Cf. A005494, A143495, A242817, A299824.

seq(add(n^(k+3)*A143495(n+3, k+3), k = 0..n), n = 0..17); # Peter Luschny, Oct 21 2020

a[n_] := Exp[-n] * Sum[j^n * n^j/(j - 3)!, {j, 3, Infinity}]; Array[a, 17, 0] (* Amiram Eldar, Oct 20 2020 *)

a(n)={sum(k=0, n+3, n^k*(stirling(n+3,k,2) - 3*stirling(n+2,k,2) + 2*stirling(n+1,k,2)))} \\ Andrew Howroyd, Oct 20 2020

# Increase precision for larger n!
R = RealField(100)
t = 3
sol = [0]*18
for n in range(0, 18):
    suma = R(0)
    for j in range(t, 1000):
        suma += (j^n * n^j) / factorial(j - t)
    suma *= exp(-n)
    sol[n] = round(suma)
print(sol) # Peter Luschny, Oct 20 2020

a(n) = Sum_{k=0..n+3} n^k*(Stirling2(n+3,k) - 3*Stirling2(n+2,k) + 2*Stirling2(n+1,k)). - Andrew Howroyd, Oct 20 2020

a(n) = Sum_{k=0..n} n^(k+3)*A143495(n+3, k+3). - Peter Luschny, Oct 21 2020

A367818 Expansion of e.g.f. exp(1 - 3*x - exp(x)).

Original entry on oeis.org

1, -4, 15, -53, 178, -575, 1809, -5598, 17141, -52113, 157724, -475997, 1433429, -4311364, 12958627, -38909601, 116831426, -350844883, 1051414421, -3160120038, 9491592177, -28218244109, 86403627444, -255153772169, 722619907385, -2772952748516, 4627276967623, -17420488524253

Offset: 0

Ilya Gutkovskiy, Dec 01 2023

sign

Cf. A000587, A005494, A109747, A153732, A193683, A346738, A367819.

nmax = 27; CoefficientList[Series[Exp[1 - 3 x - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 27}]

my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 3*x - exp(x)))) \\ Michel Marcus, Dec 02 2023

G.f. A(x) satisfies: A(x) = 1 - 3*x*A(x) - x * A(x/(1 - x)) / (1 - x).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k-3)^n / k!.
a(0) = 1; a(n) = -3*a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A000587(k).

cp's OEIS Frontend

A232473 3-Fubini numbers.

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A232474 4-Fubini numbers.

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A343673 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

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A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

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MATLAB

Mathematica

PARI

PARI

PARI

Formula

A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + 3)^n / (2^k * k!).

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Mathematica

Formula

A355163 a(n) = exp(-1) * Sum_{k>=0} (4*k + 3)^n / k!.

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Mathematica

Formula

A095674 Triangle read by rows, formed from product of Pascal's triangle (A007318) and Aitken's (or Bell's) triangle (A011971).

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Mathematica

Extensions

A282179 E.g.f.: exp(exp(x) - 1)*(exp(3*x) - 2*exp(x) + 1).

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A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2k + 3)^n / (2^k k!).

A282179 E.g.f.: exp(exp(x) - 1)(exp(3x) - 2*exp(x) + 1).