A000898 -id:A000898 - cp's OEIS frontend

A108400 a(n) = Product_{k = 0..n} (2^k * k!).

1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000

Offset: 0

Philippe Deléham, Jul 02 2005

Hankel transform (see A001906 for definition) of the sequences A000898, A001861, A035009(with first term omitted), A047974, A067147(unsigned version), A083886.
Hankel transform of the sequence with e.g.f. exp(x^2). Also (-1)^C(n+1,2)*a(n) is the Hankel transform of the sequence with e.g.f. exp(-x^2). - Paul Barry, Feb 12 2008
Let T(n,k) = (n+1)^k * (1+(-1)^(n-k))/2, then a(n) = det(T(i,j); 0<=i, j<=n). - Paul Barry, Feb 12 2008

G. C. Greubel, Table of n, a(n) for n = 0..38
M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A000178, A006125, A074962.

Cf. A000898, A001861, A001906, A035009, A047974, A067147, A083886.

BarnesG:= func< n | (&*[Factorial(j): j in [0..n-2]]) >;
[2^Binomial(n+1,2)*BarnesG(n+2): n in [0..15]]; // G. C. Greubel, Jun 21 2022

Table[Product[k!*2^k, {k,0,n}], {n,0,10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[2^Binomial[n+1,2]*BarnesG[n+2], {n,0,15}] (* G. C. Greubel, Jun 21 2022 *)

def barnes_g(n): return product(factorial(j) for j in (0..n-2))
[2^binomial(n+1,2)*barnes_g(n+2) for n in (0..15)] # G. C. Greubel, Jun 21 2022

a(n) = A006125(n+1)*A000178(n).
a(n) = Product_{i=1..n} Product_{j=0..i-1} {2*(i-j)}. - Paul Barry, Aug 02 2008
a(n) ~ 2^((n+1)^2/2) * n^(n^2/2+n+5/12) * Pi^((n+1)/2) / (A * exp(3*n^2/4+n-1/12)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

A202828 Expansion of e.g.f.: exp(4x/(1-2x)) / sqrt(1-4*x^2).

Original entry on oeis.org

1, 4, 36, 400, 5776, 97344, 1915456, 42406144, 1049760000, 28558296064, 848579961856, 27271456395264, 943132599095296, 34877026635366400, 1373536895379849216, 57351382681767706624, 2530646978003730497536, 117614221470591038521344, 5742190572014854792806400

Offset: 0

Paul D. Hanna, Dec 25 2011

a(n) is the number of good involutions of the linear Alexander quandle (Z/4nZ, 2n+1); see Ta, Thm. 5.11 and cf. A387317. - Luc Ta, Aug 26 2025

			E.g.f.: A(x) = 1 + 4*x + 36*x^2/3! + 400*x^3/3! + 5776*x^4/4! + 97344*x^5/5! +...
where A(x) = 1 + 2^2*x + 6^2*x^2/2! + 20^2*x^3/3! + 76^2*x^4/4! + 312^2*x^5/5! +...+ A000898(n)^2*x^n/n! +...

G. C. Greubel, Table of n, a(n) for n = 0..350
Lực Ta, Good involutions of twisted conjugation subquandles and Alexander quandles, arXiv:2508.16772 [math.GT], 2025.

Cf. A000898, A202827, A202829, A202831, A202833, A202835, A202836.

Cf. A387317.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-2*x))/Sqrt(1-4*x^2) ))); // G. C. Greubel, Jun 21 2022

CoefficientList[Series[Exp[4*x/(1-2*x)]/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(4*x/(1-2*x)+x*O(x^n))/sqrt(1-4*x^2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-2*k)*n!/((n-2*k)!*k!))^2}

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k)*2^k)^2}

def A202828_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
A202828_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A000898(n)^2, where the e.g.f. of A000898 is exp(2*x + x^2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-2*k) * n!/((n-2*k)!*k!) )^2.
a(n) = ( Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k) )^2. [From formula by Vladeta Jovovic in A000898].
a(n) ~ n^n*exp(2*sqrt(2*n)-1-n)*2^(n-1). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = 2*(n+1)*a(n-1) + 4*(n-1)*(n+1)*a(n-2) - 8*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013
a(n) = 2^n*A277378(n). - R. J. Mathar, Jan 20 2020

A135401 a(n) = number of permutations (p(1),p(2),p(3),...,p(n)) of (1,2,3,...n) each of which is its own inverse and is such that p(k) = n + 1 - p(n+1-k) for all k in the range 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 20, 20, 76, 76, 312, 312, 1384, 1384, 6512, 6512, 32400, 32400, 168992, 168992, 921184, 921184, 5222208, 5222208, 30710464, 30710464, 186753920, 186753920, 1171979904, 1171979904, 7573069568, 7573069568, 50305536256, 50305536256, 342949298688

Offset: 0

Leroy Quet, Dec 11 2007

nonn

a(n) is also the number of ways to place n points on an n X n grid with pairwise distinct abscissa, pairwise distinct ordinate, mirror symmetry and 180-degree rotational symmetry. Note that both diagonals are actually axes of symmetry. See also A297708, A000085, A001813, and A006882. - Manfred Scheucher, Jan 04 2018

a(n) is the number of standard Young tableaux of size n invariant under Schützenberger involution. - Ludovic Schwob, Feb 17 2024

			For n = 6 we can have the permutation (3,5,1,6,2,4). This permutation is its own inverse permutation. Furthermore, 7 = p(1)+p(6) = p(2)+p(5) = p(3)+p(4) = 3+4 = 5+2 = 1+6. So this permutation among others is included in the count of permutations when n=6.
a(4) = 6 because we have 1234, 1324, 3412, 2143, 4231 and 4321.

Manfred Scheucher, Table of n, a(n) for n = 0..99

Cf. A000085, A001813, A006882, A297708. - Manfred Scheucher, Jan 07 2018

with(combinat): with(group): a:=proc(n) local P,ct,j,pc,prc: P:=permute(n): ct:= 0: for j to factorial(n) do pc:=convert(P[j], 'disjcyc'): prc:=[seq(n+1-P[j][n+1-k],k=1..n)]: if invperm(pc)=pc and P[j]=prc then ct:=ct+1 else end if end do: ct end proc: seq(a(n),n=0..9); # Emeric Deutsch, Dec 31 2007

a898[n_] := Sum[2^k StirlingS1[n, k] BellB[k], {k, 0, n}];
a =.; a[n_] := a898[Floor[n/2]];
Table[a[n], {n, 0, 40}]
(* or: *)
a[n_] := a[n] = Which[n==0 || n==-2, 1, OddQ[n], a[n-1],
   True, 2 a[n-2] + (n-2) a[n-4]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 13 2019, updated Jul 25 2022 *)

print1(1", "1", ");a=0;b=1;for(n=1,25,c=2*(b+(n-1)*a);print1(c", "c", ");a=b;b=c) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

def a135401(n):
    return sum(binomial(n//2,2*k)*binomial(2*k,k)*factorial(k)*2**(n//2-2*k) for k in range(1+n//4))
for n in range(100): print(n, a135401(n)) # Manfred Scheucher, Jan 07 2018

a(n) = A000898(floor(n/2)). - Conjectured by Leroy Quet, Jan 20 2008; proved by Max Alekseyev, Jan 21 2008: (Start)
Let p = (p(1),...,p(n)) be a permutation such that p(k) = n + 1 - p(n+1-k) for all 1 <= k <= n.
Then 2-set {k, n+1-k} (i.e., where the sum of elements is n+1) is mapped by p into {p(k), p(n+1-k)} with the same property p(k) + p(n+1-k) = n+1. Therefore every such permutation induces a permutation q on the 2-sets {k, n+1-k}, and for odd n has a fixed point p((n+1)/2) = (n+1)/2.
Furthermore, it is easy to see that if p is self-inverse then so is q.
Let s=floor(n/2). For every permutation q on the sets {k, n+1-k}, 1 <= k <= s, let's count how many p induce it.
It is clear that if q has exactly m fixed points (and so the other s-m 2-sets form pairs of inverses under q), then there exist 2^m ways to define p on the fixed points of q and 2^((s-m)/2) ways to define p on the remaining elements.
Hence the total number of permutations p inducing q is 2^m * 2^((s-m)/2) = 2^((s+m)/2). The number of permutations q on s elements with exactly m fixed points is nonzero only if m and s are of the same oddness and in this case it is binomial(s,m) * (s-m)! / 2^((s-m)/2) / ((s-m)/2)! = binomial(s,m) * (s-m-1)!! = s! / m! / 2^((s-m)/2) / ((s-m)/2)!.
Hence a(n) = Sum s! * 2^m / m! / ((s-m)/2)!, where sum is taken over m=0,1...,s of the same oddness as s. Let s-m=2t so that m=s-2t and a(n) = Sum_{t=0..floor(s/2)} s! * 2^(s-2t) / ((s-2t)! * t!) = s! * [x^s] e^(2x) * e^(x^2) = s! * [x^s] e^(x^2+2x) = A000898(s), according to its e.g.f. QED
(End)
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2),2k) *binomial(2k,k) *k! *2^(floor(n/2)-2k). (See A000898 for the original formula by N. Calkin, for further equivalent expressions, and for (exponential) generating function.) - Manfred Scheucher, Jan 07 2018

5 more terms from Emeric Deutsch, Dec 31 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008
a(0)=1 prepended by Alois P. Heinz, Jan 03 2018

A300056 Number of normal standard domino tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 3, 1, 0, 0, 3, 0, 3, 2, 1, 0, 0, 0, 0, 1, 0, 3, 1, 0, 4, 2, 0, 0, 1, 0, 0, 1, 0, 1, 6, 0, 0, 3, 1, 0, 4, 0, 5, 0, 0, 0, 1, 1, 8, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 6, 4, 0, 0, 1, 0, 6, 1, 0, 6, 5, 0, 6, 3, 1, 2, 10, 0, 0, 1, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. A standard domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(75) = 6 tableaux:
1 2 4   1 2 3   1 2 2   1 1 4   1 1 4   1 1 3
1 2 4   1 2 3   1 3 3   2 3 4   2 2 4   2 2 3
3 3     4 4     4 4     2 3     3 3     4 4

Cf. A000085, A000712, A000898, A001222, A004003, A056239, A099390, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A297388, A299926, A300060, A300061.

A300120 Number of skew partitions whose quotient diagram is connected and whose numerator has weight n.

Original entry on oeis.org

2, 6, 12, 26, 44, 86, 136, 239, 376, 613, 930, 1485, 2194, 3355, 4948, 7372, 10656, 15660, 22359, 32308

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

			The a(3) = 12 skew partitions:
(3)/()   (3)/(1)   (3)/(2)    (3)/(3)
(21)/()  (21)/(11) (21)/(2)   (21)/(21)
(111)/() (111)/(1) (111)/(11) (111)/(111)

Cf. A000085, A000898, A006958, A138178, A238690, A259479, A259480, A297388, A299925, A299926, A300118, A300122, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

Original entry on oeis.org

1, 4, 13, 51, 183, 771, 3087, 13601, 59933, 278797, 1311719, 6453606, 32179898, 166075956, 871713213, 4704669005, 25831172649, 145260890323

Offset: 1

Gus Wiseman, Feb 25 2018

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

			The a(3) = 13 tableaux:
1 1 1   1 1 2   1 2 2   1 2 3
.
1 1   1 1   1 2   1 2   1 3
1     2     1     3     2
.
1   1   1   1
1   1   2   2
1   2   2   3

Cf. A000085, A000898, A006958, A138178, A238690, A259479, A259480, A296561, A297388, A299699, A299925, A299926, A300118, A300120, A300121, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]

A000902 Expansion of e.g.f. (1/2)(exp(2x + x^2) + 1).

Original entry on oeis.org

1, 1, 3, 10, 38, 156, 692, 3256, 16200, 84496, 460592, 2611104, 15355232, 93376960, 585989952, 3786534784, 25152768128, 171474649344, 1198143415040, 8569374206464, 62668198184448, 468111364627456, 3568287053001728

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
One more than the number of ordered pairs of minimally intersecting partitions such that p consists of exactly two blocks.
The number of B-orbits in the symmetric space of type DIII, SO_{2n}(C)/GL_n(C) where B is a Borel subgroup of SO_{2n}(C). These are parameterized by "type DIII (n,n)-clans". E.g., for n=2, the a(2)=3 type DIII (2,2)-clans are ++--, --++, and 1212. See [Bingham and Ugurlu] link. - Aram Bingham, Feb 08 2020

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
Aram Bingham and Özlem Uğurlu, Sects, rooks, pyramids, partitions and paths for type DIII clans, arXiv:1907.08875 [math.CO], 2019.
Aram Bingham and Özlem Uğurlu, DIII clan combinatorics for the orthogonal Grassmannian, Australasian J. of Combinatorics (2021) Vol. 79, No. 1, 55-86.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
Édouard Lucas, Théorie des nombres (annotated scans of a few selected pages)
B. Pittel, Where the typical set partitions meet and join, Electron. J. of Combin. 7, R5.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Equals 1/2 * A000898(n) for n>0.

a000902 n = a000902_list !! n
a000902_list = 1 : 1 : 3 : map (* 2) (zipWith (+)
   (drop 2 a000902_list) (zipWith (*) [2..] $ tail a000902_list))
-- Reinhard Zumkeller, Sep 10 2013

a:=[1,3]; [1] cat [n le 2 select a[n] else 2*Self(n-1) + (2*n-2)*Self(n-2):n in [1..22]]; // Marius A. Burtea, Feb 12 2020

# Comment from the authors: For Maple program see A000903.
A000902 := n -> `if`(n=0, 1, I^(-n)*orthopoly[H](n, I)/2):
seq(A000902(n), n=0..22); # Peter Luschny, Nov 29 2017

n = 22; CoefficientList[ Series[(1/2)*(Exp[2*x+x^2] + 1), {x, 0, n}], x] * Table[k!, {k, 0, n}]
(* Jean-François Alcover, May 18 2011 *)
With[{nn=30},CoefficientList[Series[(Exp[2x+x^2]+1)/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 27 2025 *)

a(n) = 2*a(n-1) + (2n-2)*a(n-2) for n >= 3. - N. J. A. Sloane, Sep 23 2006
a(n) = 1 + n!/(2e) * [x^n] Sum[l>=0, 1/l! * {(1+x)^l-1}^2].
For asymptotics see the Robinson paper.
But the asymptotic formula in the Robinson paper is wrong (see A000898, discussion from Oct 01 2013). - Vaclav Kotesovec, Aug 04 2014
a(n) ~ 2^(n/2-3/2) * n^(n/2) * exp(sqrt(2*n)-n/2-1/2). - Vaclav Kotesovec, Aug 04 2014
a(n) = (i/2)^(1 - n)*KummerU((1 - n)/2, 3/2, -1) for n>=1. - Peter Luschny, Nov 29 2017
a(n) = Sum_{r=0..floor(n/2)} 2^(n-2r-1) * {(n!)/(r!(n-2r)!)}. - Aram Bingham, Feb 08 2020

A192989 Expansion of e.g.f.: exp((1+x)^3 - 1).

Original entry on oeis.org

1, 3, 15, 87, 585, 4383, 35919, 318195, 3015441, 30354075, 322626159, 3603292047, 42120047385, 513557128503, 6512375759535, 85673471945067, 1166675225150241, 16413589529042355, 238151194659626319, 3558129109803374535

Offset: 0

Paul D. Hanna, Jul 13 2011

nonn

			E.g.f.: A(x) = 1 + 3*x + 15*x^2/2! + 87*x^3/3! + 585*x^4/4! +...

Seiichi Manyama, Table of n, a(n) for n = 0..561

Cf. A000110, A000898, A008275.

List([0..20], n-> Sum([0..n], k-> (-1)^(n-k)*3^k*Bell(k)* Stirling1(n,k) )); # G. C. Greubel, Jul 25 2019

[(&+[3^k*Bell(k)*StirlingFirst(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 25 2019

CoefficientList[Series[E^((1+x)^3-1), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 15 2014 *)
Table[Sum[ (-1)^(n - k) Abs[StirlingS1[n, k]] 3^k BellB[k], {k, 0, n}], {n, 0, 20}] (* Emanuele Munarini, Aug 31 2017 *)

a(n) := sum((-1)^(n-k)*abs(stirling1(n,k))*3^k*belln(k),k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 31 2017 */

{a(n)=if(n<0, 0, n!*polcoeff(exp((1+x)^3-1+x*O(x^n)), n))}

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k)*3^k)}

[sum((-1)^(n-k)*3^k*bell_number(k)*stirling_number1(n,k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 25 2019

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k) * 3^k.
Conjecture: a(n) -3*a(n-1) +6*(-n+1)*a(n-2) -3*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 12 2014
Remark: the above conjectured recurrence is true and can be easily obtained from the e.g.f. - Emanuele Munarini, Aug 31 2017
a(n) ~ 3^(n/3-1/2) * exp(-2*n/3 + 3^(1/3)*n^(2/3) + 3^(-1/3)*n^(1/3) - 2/3) * n^(2*n/3) * (1 + 23/(54*(n/3)^(1/3)) + 3149/(29160*(n/3)^(2/3))). - Vaclav Kotesovec, Jul 15 2014

A294042 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 87, 76, 1, 0, 1, 6, 45, 232, 585, 312, 1, 0, 1, 7, 66, 485, 2248, 4383, 1384, 1, 0, 1, 8, 91, 876, 6145, 24544, 35919, 6512, 1, 0, 1, 9, 120, 1435, 13716, 88245, 295456, 318195, 32400, 1, 0

Offset: 0

Seiichi Manyama, Oct 22 2017

			Square array A(n,k) begins:
   1, 1,   1,    1,     1,     1, ...
   0, 1,   2,    3,     4,     5, ...
   0, 1,   6,   15,    28,    45, ...
   0, 1,  20,   87,   232,   485, ...
   0, 1,  76,  585,  2248,  6145, ...
   0, 1, 312, 4383, 24544, 88245, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A000012, A000898, A192989, A202824, A202825.
Rows n=0..2 give A000012, A001477, A000384.
Main diagonal gives A294045.
Cf. A000110, A293991.

A(0,k) = 1 and A(n,k) = k * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1) * A(n-j,k)/(n-j)! for n > 0.

A(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * Bell(j). - Seiichi Manyama, Jan 31 2024

A300123 Number of ways to tile the diagram of the integer partition with Heinz number n using connected skew partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 4, 10, 8, 16, 8, 32, 16, 20, 8, 64, 20, 128, 16, 40, 32, 256, 16, 52, 64, 52, 32, 512, 40, 1024, 16, 80, 128, 104, 40, 2048, 256, 160, 32, 4096, 80, 8192, 64, 104, 512, 16384, 32, 272, 104

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Gus Wiseman, The a(25) = 52 connected tilings of (3,3).

Cf. A000085, A000898, A056239, A006958, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300060, A300118, A300120, A300121, A300122, A300124.

cp's OEIS Frontend

A108400 a(n) = Product_{k = 0..n} (2^k * k!).

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A202828 Expansion of e.g.f.: exp(4*x/(1-2*x)) / sqrt(1-4*x^2).

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A135401 a(n) = number of permutations (p(1),p(2),p(3),...,p(n)) of (1,2,3,...n) each of which is its own inverse and is such that p(k) = n + 1 - p(n+1-k) for all k in the range 1 <= k <= n.

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A300056 Number of normal standard domino tableaux whose shape is the integer partition with Heinz number n.

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A300120 Number of skew partitions whose quotient diagram is connected and whose numerator has weight n.

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Mathematica

A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

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Mathematica

A000902 Expansion of e.g.f. (1/2)*(exp(2*x + x^2) + 1).

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A202828 Expansion of e.g.f.: exp(4x/(1-2x)) / sqrt(1-4*x^2).

A000902 Expansion of e.g.f. (1/2)(exp(2x + x^2) + 1).