A000898 -id:A000898 - cp's OEIS frontend

A122018 Modulo 2 recursion switch between A000898 and A121966: A000898 first.

1, 2, 6, 2, 40, 32, 464, 272, 7040, 4864, 136448, 87808, 3177472, 2123776, 86861824, 57128960, 2720112640, 1806049280, 96095928320, 63587041280, 3778819358720, 2507078533120, 163724570132480, 108568842403840, 7748467910901760

Offset: 0

Roger L. Bagula, Sep 11 2006

nonn

Robert Israel, Table of n, a(n) for n = 0..731

Cf. A000898, A121966.

f:= proc(n) option remember; if n::odd then procname(n-1)-(n-1)*procname(n-2) else 2*procname(n-1)+2*(n-1)*procname(n-2) fi end proc:
f(0):= 1: f(1):= 2:
map(f, [$0..50]); # Robert Israel, Jun 20 2018

a[0] = 1; a[1] = 2; a[n_] := a[n] = If[Mod[n, 2] == 1, a[n - 1] - (n - 1)*a[n - 2], 2*(a[n - 1] + (n - 1)*a[n - 2])] b = Table[a[n], {n, 0, 30}]

For n>=2, a(n) = a(n-1) - (n-1)*a(n-2) if n is odd, a(n) = 2*a(n-1) + 2*(n-1)*a(n-2) if n is even. - Edited by Robert Israel, Jun 20 2018

Offset changed by Robert Israel, Jun 20 2018

A047974 a(n) = a(n-1) + 2(n-1)a(n-2).

Original entry on oeis.org

1, 1, 3, 7, 25, 81, 331, 1303, 5937, 26785, 133651, 669351, 3609673, 19674097, 113525595, 664400311, 4070168161, 25330978113, 163716695587, 1075631907655, 7296866339961, 50322142646161, 356790528924523, 2570964805355607, 18983329135883665, 142389639792952801, 1091556096587136051

Offset: 0

N. J. A. Sloane

nonn

Related to partially ordered sets. - Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003
The number of partial permutation matrices P in GL_n with P^2=0. Alternatively, the number of orbits of the Borel group of upper triangular matrices acting by conjugation on the set of matrices M in GL_n with M^2=0. - Brian Rothbach (rothbach(AT)math.berkeley.edu), Apr 16 2004
Number of ways to use the elements of {1..n} once each to form a collection of sequences, each having length 1 or 2. - Bob Proctor, Apr 18 2005
Hankel transform is A108400. - Paul Barry, Feb 11 2008
This is also the number of subsets of equivalent ways to arrange the elements of n pairs, when equivalence is defined under the joint operation of (optional) reversal of elements combined with permutation of the labels and the subset maps to itself. - Ross Drewe, Mar 16 2008
Equals inverse binomial transform of A000898. - Gary W. Adamson, Oct 06 2008
a(n) is also the moment of order n for the measure of density exp(-(x-1)^2/4)/(2*sqrt(Pi)) over the interval -oo..oo. - Groux Roland, Mar 26 2011
The n-th term gives the number of fixed-point-free involutions in S_n^B, the group of permutations on the set {-n,...,-1,1,2,...,n}. - Matt Watson, Jul 26 2012
From Peter Bala, Dec 03 2017: (Start)
a(n+k) == a(n) (mod k) for all n and k. Hence for each k, the sequence a(n) taken modulo k is a periodic sequence and the exact period divides k. Cf. A115329.
More generally, the same divisibility property holds for any sequence with an e.g.f. of the form F(x)*exp(x*G(x)), where F(x) and G(x) are power series with integer coefficients and G(0) = 1. See the Bala link for a proof. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Peter Alspaugh, James Garrett, Nataša Jonoska, and Masahico Saito, Structures of Monoids Motivated by DNA Origami, arXiv:2501.14966 [math.RA], 2025. See p. 22.
Tewodros Amdeberhan, Valerio De Angelis, Atul Dixit, Victor H. Moll, and Christophe Vignat, From sequences to polynomials and back, via operator orderings, J. Math. Phys. 54, 123502 (2013); Alternative copy
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts); Alternative copy
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394; Alternative copy.
Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.
Samuele Giraudo, Combalgebraic structures on decorated cliques, arXiv:1709.08416 [math.CO], 2017; and also, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8.
Tom Halverson and Mike Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150 [math.RT], 2013.
Andrei Khruzin, Enumeration of chord diagrams, arXiv:math/0008209 [math.CO], 2000.
G. Latouche and P. G. Taylor, A stochastic fluid model for an ad hoc mobile network, Queueing Syst. 63, No. 1-4, 109-129 (2009), eq. (1).
Robert A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Jocelyn Quaintance and Harris Kwong, Permutations and combinations of colored multisets, JIS 13 (2010) #10.2.6.
Index entries for related partition-counting sequences
Index entries for sequences related to Hermite polynomials

Row sums of A067147.
Column k=2 of A359762.
Cf. A000085, A000680, A000898, A001147, A001813, A100861, A108400, A115329, A132101.
Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), this sequence (q=1), A115329 (q=2), A293720 (q=4).

N = 18; A = zeros(N,1); for n = 1:N; a = factorial(n); s = 0; k = 0; while k <= floor(n/2); b = factorial(n - 2*k); c = factorial(k); s = s + a/(b*c); k = k+1; end; A(n) = s; end; disp(A); % Ross Drewe, Mar 16 2008

[n le 2 select 1 else Self(n-1) + 2*(n-2)*Self(n-2): n in [1..40]]; // G. C. Greubel, Jul 12 2024

seq( add(n!/((n-2*k)!*k!), k=0..floor(n/2)), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001
with(combstruct):seq(count(([S,{S=Set(Union(Z,Prod(Z,Z)))},labeled],size=n)),n=0..30); # Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003
A047974 := n -> I^(-n)*orthopoly[H](n, I/2):
seq(A047974(n), n=0..26); # Peter Luschny, Nov 29 2017

Range[0, 23]!*CoefficientList[ Series[ Exp[x*(1-x^2)/(1 - x)], {x, 0,23 }], x] - (* Zerinvary Lajos, Mar 23 2007 *)
Table[I^(-n)*HermiteH[n, I/2], {n, 0, 23}] - (* Alyssa Byrnes and C. Vignat, Jan 31 2013 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(x^2+x))) \\ Joerg Arndt, May 04 2013

[(-i)^n*hermite(n,i/2) for n in range(41)] # G. C. Greubel, Jul 12 2024

E.g.f.: exp(x^2+x). - Len Smiley, Dec 11 2001
Binomial transform of A001813 (with interpolated zeros). - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} C(k,n-k)*n!/k!. - Paul Barry, Mar 29 2007
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*(2k)!/k!; - Paul Barry, Feb 11 2008
G.f.: 1/(1-x-2*x^2/(1-x-4*x^2/(1-x-6*x^2/(1-x-8*x^2/(1-... (continued fraction). -Paul Barry, Apr 10 2009
E.g.f.: Q(0); Q(k) = 1+(x^2+x)/(2*k+1-(x^2+x)*(2*k+1)/((x^2+x)+(2*k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. Cf. A000085 and A115329. - Peter Bala, Dec 07 2011
a(n) ~ 2^(n/2 - 1/2)*exp(sqrt(n/2) - n/2 - 1/8)*n^(n/2). - Vaclav Kotesovec, Oct 08 2012
E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
a(n) = i^(-n)*H_{n}(i/2) with i the imaginary unit and H_{n} the Hermite polynomial of degree n. - Alyssa Byrnes and C. Vignat, Jan 31 2013
E.g.f.: -Q(0)/x where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
G.f.: 1/Q(0), where Q(k) = 1 + x*2*k - x/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013
E.g.f.: E(0)-1-x-x^2, where E(k) = 2 + 2*x*(1+x) - 8*k^2 + x^2*(1+x)^2*(2*k+3)*(2*k-1)/E(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013
E.g.f.: Product_{k>=1} 1/(1 + (-x)^k)^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019
a(n) = Sum_{k=0..floor(n/2)} 2^k*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

A300121 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 4, 11, 12, 16, 12, 32, 28, 31, 8, 64, 31, 128, 33, 82, 64, 256, 28, 69, 144, 69, 86, 512, 105, 1024, 16, 208, 320, 209, 82, 2048, 704, 512, 86, 4096, 318, 8192, 216, 262, 1536, 16384, 64, 465, 262, 1232, 528, 32768, 209, 588, 245, 2912, 3328

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. 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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 11 tableaux:
1 1
1 1
.
2 1   1 1   1 1   1 2
1 1   1 2   2 2   1 2
.
1 1   1 2   1 2   1 3
2 3   1 3   3 3   2 3
.
1 2   1 3
3 4   2 4

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300056, A300060, A300118, A300120, 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]]Table[PrimePi[p],{k}]]]];
Table[Length[cos[Reverse[primeMS[n]]]],{n,50}]

A000899 Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).

Original entry on oeis.org

0, 0, 0, 1, 9, 70, 571, 4820, 44676, 450824, 4980274, 59834748, 778230060, 10896609768, 163456629604, 2615335902176, 44460874280032, 800296440705472, 15205636325496568, 304112744618157872, 6386367741011250672

Offset: 1

N. J. A. Sloane

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 = 1..200
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]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Cf. A000900.

Maple
```
For Maple program see A000903.
```

a[n_] := ((n+1)! - (2*Floor[(n+1)/2])!! - 2*Sum[Binomial[n+1, 2*k]*(2*k-1)!!, {k, 0, (n+1)/2}] + 2*Sum[2^k*BellB[k]*StirlingS1[Floor[(n+1)/2], k], {k, 0, Floor[(n+1)/2]}])/8; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 23 2013, from explicit formulas *)

a(n)=(A000142(n)-2*A000085(n)-A037223(n)+2*A000898(floor(n/2)))/8 (all of which have explicit formulas).

For asymptotics see the Robinson paper.

More terms from Vladeta Jovovic, May 09 2000

A299968 Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 189, 753, 3248, 14738, 70658, 354178, 1857703, 10121033, 57224955, 334321008, 2017234773, 12530668585, 80083779383, 525284893144, 3533663143981, 24336720018666, 171484380988738, 1234596183001927, 9075879776056533, 68052896425955296

Offset: 0

Gus Wiseman, Feb 26 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.

			The a(4) = 15 tableaux:
1 2 3 4
.
1 2 3   1 2 4   1 3 4   1 2 3   1 2 3
4       3       2       2       3
.
1 2   1 3   1 2
3 4   2 4   2 3
.
1 2   1 3   1 2   1 4   1 3
3     2     2     2     2
4     4     3     3     3
.
1
2
3
4

Alois P. Heinz, Table of n, a(n) for n = 0..50 (first 46 terms from Ludovic Schwob)
D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific Journal of Mathematics, Vol. 34, No. 3 (1970), 709-727.
Wikipedia, Young tableau

Cf. A000085, A000898, A001221, A005117, A006958, A015128, A138178, A238121, A238690, A285175, A296561, A297388, A299926, A300120, A300122.

unddis[y_]:=DeleteCases[y-#,0]&/@Tuples[Table[If[y[[i]]>Append[y,0][[i+1]],{0,1},{0}],{i,Length[y]}]];
dos[y_]:=With[{sam=Rest[unddis[y]]},If[Length[sam]===0,If[Total[y]===0,{{}},{}],Join@@Table[Prepend[#,y]&/@dos[sam[[k]]],{k,1,Length[sam]}]]];
Table[Sum[Length[dos[y]],{y,IntegerPartitions[n]}],{n,1,8}]

a(n) = Sum_{k=0..n} 2^k * A238121(n,k). - Ludovic Schwob, Sep 23 2023

More terms from Ludovic Schwob, Sep 23 2023

A062267 Row sums of (signed) triangle A060821 (Hermite polynomials).

Original entry on oeis.org

1, 2, 2, -4, -20, -8, 184, 464, -1648, -10720, 8224, 230848, 280768, -4978816, -17257600, 104891648, 727511296, -1901510144, -28538404352, 11377556480, 1107214478336, 1759326697472, -42984354695168, -163379084079104

Offset: 0

Wolfdieter Lang, Jun 19 2001

G. C. Greubel, Table of n, a(n) for n = 0..730
Index entries for sequences related to Hermite polynomials

Cf. A000898, A121966.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(2-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 08 2018

A062267 := proc(n)
    HermiteH(n,1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Feb 05 2013

lst={};Do[p=HermiteH[n,1];AppendTo[lst,p],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Table[2^n HypergeometricU[-n/2, 1/2, 1], {n, 0, 23}] (* Benedict W. J. Irwin, Oct 17 2017 *)
With[{nmax=50}, CoefficientList[Series[Exp[x*(2-x)], {x,0,nmax}],x]* Range[0, nmax]!] (* G. C. Greubel, Jun 08 2018 *)

x='x+O('x^30); Vec(serlaplace(exp(-x*(x-2)))) \\ G. C. Greubel, Jun 08 2018

a(n) = polhermite(n,1); \\ Michel Marcus, Jun 09 2018

from sympy import hermite, Poly
def a(n): return sum(Poly(hermite(n, x), x).all_coeffs()) # Indranil Ghosh, May 26 2017

a(n) = Sum_{m=0..n} A060821(n, m) = H(n, 1), with the Hermite polynomials H(n, x).
E.g.f.: exp(-x*(x-2)).
a(n) = 2*(a(n - 1) - (n - 1)*a(n - 2)). - Roger L. Bagula, Sep 11 2006
a(n) = 2^n * U(-n/2, 1/2, 1), where U is the confluent hypergeometric function. - Benedict W. J. Irwin, Oct 17 2017
E.g.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

Original entry on oeis.org

1, 1, 2, 7, 23, 115, 694, 5282, 46066, 456454, 4999004, 59916028, 778525516, 10897964660, 163461964024, 2615361578344, 44460982752488, 800296985768776, 15205638776753680, 304112757426239984, 6386367801916347184

Offset: 1

N. J. A. Sloane

			For n=4 the 7 solutions may be taken to be 1234,1243,1324,1423,1432,2143,2413.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. C. Read, personal communication.
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).
Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

N. J. A. Sloane, Table of n, a(n) for n = 1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
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]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 9.
Eric Weisstein's World of Mathematics, Rook Number.
Eric Weisstein's World of Mathematics, Rooks Problem.

Cf. A000142, A037223, A037224, A000085, A005635, A099952, A263685.

Maple programs for A000142, A037223, A122670, A001813, A000085, A000898, A000407, A000902, A000900, A000901, A000899, A000903
P:=n->n!; # Gives A000142
G:=proc(n) local k; k:=floor(n/2); k!*2^k; end; # Gives A037223, A000165
R:=proc(n) local m; if n mod 4 = 2 or n mod 4 = 3 then RETURN(0); fi; m:=floor(n/4); (2*m)!/m!; end; # Gives A122670, A001813
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
B:=proc(n) option remember; if n <= 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(B(n-1)); fi; 2*B(n-2) + (n-2)*B(n-4); end; # Gives A000898 (doubled up)
rho:=n->R(n)/2; # Gives A000407, aerated
beta:=n->B(n)/2; # Gives A000902, doubled up
delta:=n->(D(n)-B(n))/2; # Gives A000900
unprotect(gamma); gamma:=n-> if n <= 1 then RETURN(0) else (G(n)-B(n)-R(n))/4; fi; # Gives A000901, doubled up
alpha:=n->P(n)/8-G(n)/8+B(n)/4-D(n)/4; # Gives A000899
unprotect(sigma); sigma:=n-> if n <= 1 then RETURN(1); else P(n)/8+G(n)/8+R(n)/4+D(n)/4; fi; #Gives A000903

c[n_] := Floor[n/2]! 2^Floor[n/2];
r[n_] := If[Mod[n, 4] > 1, 0, m = Floor[n/4]; If[m == 0, 1, (2 m)!/m!]];
d[0] = d[1] = 1; d[n_] := d[n] = (n - 1)d[n - 2] + d[n - 1];
a[1] = 1; a[n_] := (n! + c[n] + 2 r[n] + 2 d[n])/8;
Array[a, 21] (* Jean-François Alcover, Apr 06 2011, after Matthias Engelhardt, further improved by Robert G. Wilson v *)

If n>1 then a(n) = 1/8 * (F(n) + C(n) + 2 * R(n) + 2 * D(n)), where F(n) = A000142(n) [all solutions, i.e., factorials], C(n) = A037223(n) [central symmetric solutions], R(n) = A037224(n) [rotationally symmetric solutions] and D(n) = A000085(n) [symmetric solutions by reflection at a diagonal]. - Matthias Engelhardt, Apr 05 2000

For asymptotics see the Robinson paper.

More terms from David W. Wilson, Jul 13 2003

A000900 Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).

Original entry on oeis.org

0, 0, 0, 1, 2, 10, 28, 106, 344, 1272, 4592, 17692, 69384, 283560, 1191984, 5171512, 23087168, 105883456, 498572416, 2404766224, 11878871456, 59975885856, 309439708352, 1628919330208, 8746079933568, 47840206525056

Offset: 0

N. J. A. Sloane

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
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]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
R. G. Wilson, v, Comments on the Larsen paper (no date)

Maple
```
For Maple program see A000903.
```

a85[n_] := Sum[ (2k)!/k!/2^k Binomial[n, 2k], {k, 0, n/2}]; a898[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; a[n_] := (a85[n] - a898[Floor[n/2]])/2; a[1] = 0; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 13 2011, after formula *)

a(n)=(A000085(n)-A000898(int(n/2)))/2

For asymptotics see the Robinson paper.

More terms from Vladeta Jovovic, May 09 2000

A300788 Number of strict integer partitions of n in which the even parts appear as often at even positions as at odd positions.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 19, 23, 26, 30, 35, 42, 47, 54, 62, 73, 82, 94, 107, 124, 139, 158, 179, 206, 230, 260, 293, 334, 372, 420, 470, 532, 591, 664, 740, 835, 924, 1034, 1148, 1288, 1422, 1588, 1756, 1962, 2161, 2404

Offset: 0

Gus Wiseman, Mar 12 2018

nonn

			The a(9) = 3 strict partitions: (9), (621), (531). Missing are: (81), (72), (63), (54), (432).

Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 291 terms from Fausto A. C. Cariboni)

Cf. A000712, A000898, A001405, A026010, A045931, A063886, A097613, A130780, A171966, A239241, A300787, A300789.

cobal[y_]:=Sum[(-1)^x,{x,Join@@Position[y,_?EvenQ]}];
Table[Length[Select[IntegerPartitions[n],cobal[#]===0&&UnsameQ@@#&]],{n,0,40}]

a(41)-a(58) from Alois P. Heinz, Mar 13 2018

A300060 Number of domino tilings of the diagram of 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, 2, 1, 0, 0, 3, 0, 3, 1, 1, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 5, 0, 0, 1, 1, 0, 3, 0, 2, 0, 0, 0, 1, 1, 3, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 4, 1, 0, 0, 1, 0, 5, 1, 0, 2, 3, 0, 2, 1, 1, 1, 5, 0, 0, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Alois P. Heinz, Table of n, a(n) for n = 1..100000
Wikipedia, Domino tiling
Gus Wiseman, The a(91) = 5 domino tilings of (6,4).

Cf. A000085, A000720, A000712, A000898, A001222, A004003, A056239, A099390, A138178, A153452, A238690, A296150, A296188, A299925, A299926, A300056, A300061, A304662.

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
      fi
    end:
g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
        `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
a:= n-> g(sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..120);  # Alois P. Heinz, May 22 2018

h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[Map[Function[x, x-1], l[[Range @ f[[1]]]]], ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]] > 0, k-- ]; If[Length[f] > 0 && f[[1]] >= k, h[ReplacePart[l, k -> 2], f], 0] + If[k > 1 && l[[k-1]] == 0, h[ReplacePart[l, {k -> 1, k - 1 -> 1}], f], 0]]];
g[l_] := If[Sum[If[OddQ @ l[[i]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, l[[1]]], ReplacePart[l, 1 -> Nothing]]], 0];
a[n_] := g[Reverse @ Sort[ Flatten[ Map[ Function[i, Table[PrimePi[i[[1]]], i[[2]]]], FactorInteger[n]]]]];
Array[a, 120] (* Jean-François Alcover, May 28 2018, after Alois P. Heinz *)

cp's OEIS Frontend

A122018 Modulo 2 recursion switch between A000898 and A121966: A000898 first.

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A047974 a(n) = a(n-1) + 2*(n-1)*a(n-2).

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A300121 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.

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A000899 Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).

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A299968 Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.

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A062267 Row sums of (signed) triangle A060821 (Hermite polynomials).

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A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

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A047974 a(n) = a(n-1) + 2(n-1)a(n-2).