A134980 -id:A134980 - cp's OEIS frontend

A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 10, 4, 1, 52, 52, 37, 17, 5, 1, 203, 203, 151, 77, 26, 6, 1, 877, 877, 674, 372, 141, 37, 7, 1, 4140, 4140, 3263, 1915, 799, 235, 50, 8, 1, 21147, 21147, 17007, 10481, 4736, 1540, 365, 65, 9, 1, 115975, 115975, 94828, 60814, 29371, 10427, 2727, 537, 82, 10, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

The column for k=0 is A000110 (Bell or exponential numbers). The column for k=1 is A000110 starting at offset 1. The column for k=2 is A005493 (Sum_{k=0..n} k*Stirling2(n,k).). The column for k=3 is A005494 (E.g.f.: exp(3*z+exp(z)-1).). The column for k=4 is A045379 (E.g.f.: exp(4*z+exp(z)-1).). The row for n=0 is 1's sequence, the row for n=1 is the natural numbers. The row for n=2 is A002522 (n^2 + 1.). The row for n=3 is A005491 (n^3 + 3n + 1.). The row for n=4 is A005492.
Number of ways of placing n labeled balls into n+k boxes, where k of the boxes are labeled and the rest are indistinguishable. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
The column for k = -1 (not shown) is A000296 (Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.). - Gerald McGarvey, Oct 08 2006
Equals antidiagonals of an array in which (n+1)-th column is the binomial transform of n-th column, with leftmost column = the Bell sequence, A000110. - Gary W. Adamson, Apr 16 2009
Number of partitions of [n+k] where at least k blocks contain their own index element.  A(2,2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - Alois P. Heinz, Jan 07 2022

			Array A(n,k) begins:
   1,   1,   1,    1,    1,     1,     1,     1,     1,      1, ... A000012;
   1,   2,   3,    4,    5,     6,     7,     8,     9,     10, ... A000027;
   2,   5,  10,   17,   26,    37,    50,    65,    82,    101, ... A002522;
   5,  15,  37,   77,  141,   235,   365,   537,   757,   1031, ... A005491;
  15,  52, 151,  372,  799,  1540,  2727,  4516,  7087,  10644, ... A005492;
  52, 203, 674, 1915, 4736, 10427, 20878, 38699, 67340, 111211, ... ;
Antidiagonal triangle, T(n, k), begins as:
     1;
     1,    1;
     2,    2,    1;
     5,    5,    3,    1;
    15,   15,   10,    4,   1;
    52,   52,   37,   17,   5,   1;
   203,  203,  151,   77,  26,   6,  1;
   877,  877,  674,  372, 141,  37,  7,  1;
  4140, 4140, 3263, 1915, 799, 235, 50,  8,  1;

F. Ruskey, Combinatorial Generation, preprint, 2001.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1, Figure 1.
J. Riordan, Letter, Oct 31 1977, The array is on the second page.
F. Ruskey, Combinatorial Generation, 2003.
F. Ruskey, Lexicographic Algorithms [Broken link]

Cf. A000012, A000027, A000110, A002522, A005490, A005491.
Cf. A005492, A005493, A005494, A045379, A086659, A124824.
Main diagonal gives A134980.
Antidiagonal sums give A347420.

A108087:= func< n,k | (&+[Binomial(n-k,j)*k^j*Bell(n-k-j): j in [0..n-k]]) >;
[A108087(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 02 2022

with(combinat):
A:= (n, k)-> add(binomial(n, i) * k^i * bell(n-i), i=0..n):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 18 2012

Unprotect[Power]; 0^0 = 1; A[n_, k_] := Sum[Binomial[n, i] * k^i * BellB[n - i], {i, 0, n}]; Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

f(n,k)=round (suminf(i=0,(i+k)^n/i!)/exp(1));
g(n,k)=for(k=0,k,print1(f(n,k),",")) \\ prints k+1 terms of n-th row

def A108087(n,k): return sum( k^j*bell_number(n-k-j)*binomial(n-k,j) for j in range(n-k+1))
flatten([[A108087(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 02 2022

For n> 1, A(n, k) = k^n + sum_{i=0..n-2} A086659(n, i)*k^i. (A086659 is set partitions of n containing k-1 blocks of length 1, with e.g.f: exp(x*y)*(exp(exp(x)-1-x)-1).)
A(n, k) = k * A(n-1, k) + A(n-1, k+1), A(0, k) = 1. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
A(n,k) = Sum_{i=0..n} C(n,i) * k^i * Bell(n-i). - Alois P. Heinz, Jul 18 2012
Sum_{k=0..n-1} A(n-k,k) = A005490(n). - Alois P. Heinz, Jan 05 2022
From G. C. Greubel, Dec 02 2022: (Start)
T(n, n) = A000012(n).
T(n, n-1) = A000027(n).
T(n, n-2) = A002522(n-1).
T(n, n-3) = A005491(n-2).
T(n, n-4) = A005492(n+1).
T(2*n, n) = A134980(n).
T(2*n, n+1) = A124824(n), n >= 1.
Sum_{k=0..n} T(n, k) = A347420(n). (End)

A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).

Original entry on oeis.org

1, 0, 2, -13, 127, -1573, 23711, -421356, 8626668, -199971255, 5177291275, -148078588667, 4636966634653, -157786054331852, 5797411243015250, -228749440644895405, 9646951350227609155, -433035586385769361001, 20614401475233006857035, -1037331650810058231498688

Offset: 0

Ilya Gutkovskiy, Oct 06 2017

sign

The n-th term of the n-th inverse binomial transform of A000110.

Seiichi Manyama, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Cf. A000110, A000296, A126617, A134980, A346738, A346739, A346740.

Main diagonal of A361781.

R:=PowerSeriesRing(Rationals(), 50);
A290219:= func< n | Coefficient(R!(Laplace( Exp(Exp(x)-n*x-1) )), n) >;
[A290219(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

b:= proc(n, k) option remember; `if`(n=0, 1,
      k*b(n-1, k)+ b(n-1, k+1))
    end:
a:= n-> b(n, -n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

Table[n! SeriesCoefficient[Exp[Exp[x] - n x - 1], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n, k] BellB[k] , {k, 0, n}], {n, 1, 19}]]

[factorial(n)*( exp(exp(x) -n*x -1) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

a(n) ~ (-1)^n * exp(exp(-1) - 1) * n^n. - Vaclav Kotesovec, Aug 04 2021

A334242 a(n) = exp(-n) * Sum_{k>=0} (k + n)^n * n^k / k!.

Original entry on oeis.org

1, 2, 18, 273, 5812, 159255, 5336322, 211385076, 9663571400, 500742188415, 29002424377110, 1856728690107027, 130194428384173116, 9923500366931329282, 816909605562423271178, 72231668379957026776065, 6827368666949651984215824, 686970682778467688690704639

Offset: 0

Ilya Gutkovskiy, Apr 19 2020

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A134980, A242817, A334240, A334241, A334243.

Table[n! SeriesCoefficient[Exp[n (Exp[x] + x - 1)], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[k, n] n^(n - k), {k, 0, n}], {n, 1, 17}]]

a(n) = n! * [x^n] exp(n*(exp(x) + x - 1)).
a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(n) * n^(n-k).
a(n) ~ c * exp((r^2/(1-r) - 1)*n) * n^n / (1-r)^n, where r = A333761 = 0.59894186245845296434937... is the root of the equation LambertW(r) = 1-r and c = 0.897950293373062982395233981707095204244165706668136925178217032608352851... - Vaclav Kotesovec, Jun 09 2020

A307066 a(n) = exp(-1) * Sum_{k>=0} (n*k + 1)^n/k!.

Original entry on oeis.org

1, 2, 13, 199, 5329, 216151, 12211597, 909102342, 85761187393, 9957171535975, 1390946372509101, 229587693339867567, 44117901231194922193, 9748599124579281064294, 2451233017637221706477037, 695088863051920283838281851, 220558203335628758134165860609

Offset: 0

Ilya Gutkovskiy, Jun 24 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000110, A126390, A134980, A284859, A285064, A292914, A307080.

A307066:= func< n | (&+[Binomial(n,k)*n^k*Bell(k): k in [0..n]]) >;
[A307066(n): n in [0..31]]; // G. C. Greubel, Jan 24 2024

Table[Exp[-1] Sum[(n k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 16}]
Table[n! SeriesCoefficient[Exp[Exp[n x] + x - 1], {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[Binomial[n, k] n^k BellB[k], {k, 0, n}], {n, 1, 16}]]

def A307066(n): return sum(binomial(n,k)*n^k*bell_number(k) for k in range(n+1))
[A307066(n) for n in range(31)] # G. C. Greubel, Jan 24 2024

a(n) = n! * [x^n] exp(exp(n*x) + x - 1).

a(n) = Sum_{k=0..n} binomial(n,k) * n^k * Bell(k).

A298373 a(n) = n! * [x^n] exp(n*x - exp(x) + 1).

Original entry on oeis.org

1, 0, 0, 1, 17, 273, 4779, 93532, 2047730, 49854795, 1339872113, 39462731031, 1265248227869, 43895994373580, 1639148060192408, 65568985769784897, 2797922570156143597, 126880981472647625557, 6094210606862471240855, 309087628703330034215088, 16508178701980033054460042

Offset: 0

Ilya Gutkovskiy, Jan 18 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Cf. A000587, A074051, A134980, A193683, A193684, A196835, A290219.

R:=PowerSeriesRing(Rationals(), 50);
A298373:= func< n | Coefficient(R!(Laplace( Exp(-Exp(x)+n*x+1) )), n) >;
[A298373(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

b:= proc(n, k) option remember; `if`(n=0, 1,
      k*b(n-1, k)+ b(n-1, k-1))
    end:
a:= n-> abs(b(n, -n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

Table[n! SeriesCoefficient[Exp[n x - Exp[x] + 1], {x,0,n}], {n,0,20}]
Join[{1}, Table[Sum[Binomial[n, k] n^(n-k) BellB[k,-1] , {k,0,n}], {n,20}]]

[factorial(n)*( exp(-exp(x) +n*x+1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000587(k).

a(n) ~ exp(1-exp(1)) * n^n. - Vaclav Kotesovec, Aug 04 2021

A340822 a(n) = exp(-1) * Sum_{k>=0} (k*(k + n))^n / k!.

Original entry on oeis.org

1, 3, 43, 1211, 54812, 3572775, 313493737, 35368945463, 4962511954307, 844198388785291, 170675800745636572, 40352181663578992883, 11008690527354504977193, 3426969405868832970281647, 1205708016597226199323015459, 475502109963529414669658708847

Offset: 0

Ilya Gutkovskiy, Jan 22 2021

nonn

Cf. A000110, A020557, A094577, A134980, A334242, A340823.

Table[Exp[-1] Sum[(k (k + n))^n/k!, {k, 0, Infinity}], {n, 0, 15}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[2 n - k] n^k, {k, 0, n}], {n, 1, 15}]]

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(2*n-k) * n^k.

A363429 Number of set partitions of [n] such that each block has at most one even element.

Original entry on oeis.org

1, 1, 2, 5, 10, 37, 77, 372, 799, 4736, 10427, 73013, 163967, 1322035, 3017562, 27499083, 63625324, 646147067, 1512354975, 16926317722, 40012800675, 489109544320, 1166271373797, 15455199988077, 37134022033885, 530149003318273, 1282405154139046, 19619325078384593

Offset: 0

Alois P. Heinz, Jun 01 2023

nonn

			a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.
a(4) = 10: 123|4, 12|34, 12|3|4, 134|2, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..772
Wikipedia, Partition of a set

Bisection gives: A134980 (even part).

Cf. A000110, A110132 (exactly one even), A124421 (at least one even), A363430 (at most one odd).

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m+1)+m*b(n-1, m))
    end:
a:= n-> (h-> b(n-h, h))(iquo(n, 2)):
seq(a(n), n=0..30);

a(n) = Sum_{k=0..ceiling(n/2)} floor(n/2)^k * binomial(ceiling(n/2),k) * Bell(ceiling(n/2)-k).

A363430 Number of set partitions of [n] such that each block has at most one odd element.

Original entry on oeis.org

1, 1, 2, 3, 10, 17, 77, 141, 799, 1540, 10427, 20878, 163967, 338233, 3017562, 6376149, 63625324, 137144475, 1512354975, 3315122947, 40012800675, 88981537570, 1166271373797, 2626214876310, 37134022033885, 84540738911653, 1282405154139046, 2948058074576995

Offset: 0

Alois P. Heinz, Jun 01 2023

nonn

			a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 3: 12|3, 1|23, 1|2|3.
a(4) = 10: 124|3, 12|34, 12|3|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
a(5) = 17: 124|3|5, 12|34|5, 12|3|45, 12|3|4|5, 14|23|5, 1|234|5, 1|23|45, 1|23|4|5, 14|25|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|25|34, 1|2|34|5, 1|25|3|4, 1|2|3|45, 1|2|3|4|5.

Alois P. Heinz, Table of n, a(n) for n = 0..773
Wikipedia, Partition of a set

Bisection gives: A134980 (even part).

Cf. A000110, A110138 (exactly one odd), A124423 (at least one odd), A363429 (at most one even).

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m+1)+m*b(n-1, m))
    end:
a:= n-> (h-> b(h, n-h))(iquo(n, 2)):
seq(a(n), n=0..30);

a(n) = Sum_{k=0..floor(n/2)} ceiling(n/2)^k * binomial(floor(n/2),k) * Bell(floor(n/2)-k).

A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2k + n)^n / (2^k k!).

Original entry on oeis.org

1, 2, 11, 92, 1025, 14232, 236403, 4568720, 100670529, 2490511776, 68341981051, 2059882505408, 67645498798721, 2403948686290816, 91914992104815459, 3762299973887526144, 164148252324092964993, 7604537914425558921728, 372812121514187124192875

Offset: 0

Ilya Gutkovskiy, Aug 11 2020

nonn

Cf. A004211, A007405, A134980, A337010, A337011.

Table[n! SeriesCoefficient[Exp[n x + (Exp[2 x] - 1)/2], {x, 0, n}], {n, 0, 18}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] n^(n - k) 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 18}]

a(n) = n! * [x^n] exp(n*x + (exp(2*x) - 1) / 2).

a(n) = Sum_{k=0..n} binomial(n,k) * n^(n-k) * A004211(k).

cp's OEIS Frontend

A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.

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A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).

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A334242 a(n) = exp(-n) * Sum_{k>=0} (k + n)^n * n^k / k!.

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A307066 a(n) = exp(-1) * Sum_{k>=0} (n*k + 1)^n/k!.

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A298373 a(n) = n! * [x^n] exp(n*x - exp(x) + 1).

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A340822 a(n) = exp(-1) * Sum_{k>=0} (k*(k + n))^n / k!.

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A363429 Number of set partitions of [n] such that each block has at most one even element.

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A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2k + n)^n / (2^k k!).