A002416 -id:A002416 - cp's OEIS frontend

A273461 Number of physically stable n X n placements of water source-blocks in Minecraft.

1, 2, 9, 40, 484, 9717, 338724, 21624680, 2504301849, 520443847520, 195145309791364, 131850659243316222, 160668896658179472676, 352891729183598844656996, 1397187513066371784602204416, 9972288382286063615850619475640

Offset: 0

Gus Wiseman, May 23 2016

nonn

In Minecraft worlds, a source block of water can be reacted with another source block, two blocks away. This reaction creates a third "infinite" source block in the unoccupied intermediate block, so called because if the intermediate water source is destroyed or picked up by a player using a bucket, it will immediately regenerate itself.
A placement of water at several positions in an n X n board is said to be *stable* if no infinite water physics can in fact occur (under otherwise optimal conditions). This means that the total quantity of water in the system is held constant.
In short, no two source blocks can be graph-distance 2 from each other. - Gus Wiseman, Nov 27 2019
Often incorrectly described as cellular automata, the observed behaviors of liquids within a board are inseparable in certain ways from states of affair outside of the board and events outside of the system. This aspect of Minecraft is poorly understood.

			a(2) = 9: {{}, {(2,2)}, {(2,1)}, {(2,1),(2,2)}, {(1,2)}, {(1,2),(2,2)}, {(1,1)}, {(1,1),(2,1)}, {(1,1),(1,2)}}.

EthosLab, Minecraft - Tutorial: Water
Wikipedia, Distance (graph theory)
Gus Wiseman, Example of a physically stable arrangement of water source-blocks (n=11)
Christopher Cormier, C# Program

The one-dimensional version is A006498.
Dominated by A329871.
Cf. A002416, A005251, A027624, A114901.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
allflows[n_]:=stableSets[Join@@Array[List,{n,n}],Function[{v,w},Plus@@Abs/@(w-v)===2]];
Table[Length[allflows[i]],{i,6}] (* Gus Wiseman, May 23 2016 *)

a(7) from Tae Lim Kook, May 25 2016
a(8) from Tae Lim Kook, May 29 2016
a(7)-a(8) corrected by Christopher Cormier, Dec 17 2019
a(9)-a(15) from Christopher Cormier, Dec 19 2019

A274805 The logarithmic transform of sigma(n).

Original entry on oeis.org

1, 2, -3, -6, 45, 11, -1372, 4298, 59244, -573463, -2432023, 75984243, -136498141, -10881169822, 100704750342, 1514280063802, -36086469752977, -102642110690866, 11883894518252419, -77863424962770751, -3705485804176583500, 71306510264347489177

Offset: 1

Johannes W. Meijer, Jul 27 2016

sign

The logarithmic transform [LOG] transforms an input sequence b(n) into the output sequence a(n). The LOG transform is the inverse of the exponential transform [EXP], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell’s formula. For information about the EXP transform see A274804. The logarithmic transform is related to the inverse multinomial transform, see A274844 and the first formula.
The definition of the LOG transform, see the second formula, shows that n >= 1. To preserve the identity EXP[LOG[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the LOG transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the cumulant expansion numbers A127671 appear.
We observe that the logarithmic transform leaves the value of a(0) undefined.
The Maple programs can be used to generate the logarithmic transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A001187 and the first formula. The second program uses the definition of the logarithmic transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the logarithmic transform, see A274804.

			Some a(n) formulas, see A127671:
a(0) = undefined
a(1) = 1*x(1)
a(2) = 1*x(2) - x(1)^2
a(3) = 1*x(3) - 3*x(1)*x(2) + 2*x(1)^3
a(4) = 1*x(4) - 4*x(1)*x(3) - 3*x(2)^2 + 12*x(1)^2*x(2) - 6*x(1)^4
a(5) = 1*x(5) - 5*x(1)*x(4) - 10*x(2)*x(3) + 20*x(1)^2*x(3) + 30*x(1)*x(2)^2 - 60*x(1)^3*x(2) + 24*x(1)^5

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 1..451
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 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]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Logarithmic Transform.

Some LOG transform pairs are, n >= 1: A006125(n-1) and A033678(n); A006125(n) and A001187(n); A006125(n+1) and A062740(n); A000045(n) and A112005(n); A000045(n+1) and A007553(n); A000040(n) and A007447(n); A000051(n) and (-1)*A263968(n-1); A002416(n) and A062738(n); A000290(n) and A033464(n-1); A029725(n-1) and A116652(n-1); A052332(n) and A002031(n+1); A027641(n)/A027642(n) and (-1)*A060054(n+1)/(A075180(n-1).
Cf. A274804, A274844, A127671, A000203.
Cf. A112005, A007553, A062740, A007447, A062738, A033464, A116652, A002031, A003704, A003707, A155585, A000142, A226968.

nmax:=22: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; b(n) - add(k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n: end: seq(a(n), n=1..nmax); # End first LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: t1 := log(1 + add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=1..nmax); # End second LOG program.
nmax:=22: with(numtheory): b := proc(n): sigma(n) end: f := series(exp(add(r(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(1):=b(1): r(1):= b(1): for n from 2 to nmax+1 do r(n) := solve(d(n)-b(n), r(n)): a(n):=r(n): od: seq(a(n), n=1..nmax); # End third LOG program.

a[1] = 1; a[n_] := a[n] = DivisorSigma[1, n] - Sum[k*Binomial[n, k] * DivisorSigma[1, n-k]*a[k], {k, 1, n-1}]/n; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Feb 27 2017 *)

N=33; x='x+O('x^N); Vec(serlaplace(log(1+sum(n=1,N,sigma(n)*x^n/n!)))) \\ Joerg Arndt, Feb 27 2017

a(n) = b(n) - Sum_{k = 1..n-1}((k*binomial(n, k)*b(n-k)*a(k))/n), n >= 1, with b(n) = A000203(n) = sigma(n).

E.g.f. log(1+ Sum_{n >= 1}(b(n)*x^n/n!)), n >= 1, with b(n) = A000203(n) = sigma(n).

A283500 Triangle read by rows: T(n,k) = number of n X n (0,1) matrices with at most k 1's in each row or column.

Original entry on oeis.org

2, 7, 16, 34, 265, 512, 209, 7343, 41503, 65536, 1546, 304186, 6474726, 24997921, 33554432, 13327, 17525812, 1709852332, 21252557377, 57366997447, 68719476736, 130922, 1336221251, 702998475376, 34215495252681, 252540841305558, 505874809287625

Offset: 1

R. J. Mathar, Mar 09 2017

			Triangle begins:
2;
7,     16;
34,    265,      512;
209,   7343,     41503,      65536;
1546,  304186,   6474726,    24997921,    33554432;
13327, 17525812, 1709852332, 21252557377, 57366997447, 68719476736;
...

Andrew Howroyd, Table of n, a(n) for n = 1..84

Cf. A002720 (column k=1), A197458 (column k=2), A008300 (exactly k 1s).
Main diagonal and first lower diagonal give: A002416, A048291.
Cf. A247158 (k=n/2).

A329871 Number of static n X n placements of water source-blocks in Minecraft.

Original entry on oeis.org

1, 2, 10, 55, 754, 18853, 82931, 70143802, 11087020614, 3243227117597, 1772826333285009, 1806938280429412270, 3430002591378184399879, 12137184871791092506807847, 80047171080361800628780500638, 983838070049011459232146327319193

Offset: 0

Gus Wiseman, Nov 26 2019

nonn

In Minecraft worlds, a source block of water can be reacted with another source block, two blocks away, linearly or diagonally. This reaction creates a third "infinite" source block in the unoccupied intermediate block or blocks, so called because if the intermediate water source is destroyed or picked up by a player using a bucket, it will immediately regenerate itself.

A placement of water at several positions in an n X n board is said to be static if no infinite water sources are created that are not already present. In particular, the total quantity of water in the system is held constant.

EthosLab, Minecraft - Tutorial: Water
Gus Wiseman, The a(3) = 55 static placements of water source-blocks (black = water).
Christopher Cormier, C# Program

Dominates A273461.
The one-dimensional case is A005251.
Cf. A002416, A006498, A027624, A114901.

vdist[v_,w_]:=Total[Abs[v-w]];
flowdown[prs_]:=Union[prs,With[{ovs=Select[Subsets[prs,{2}],vdist@@#==2&]},Union@@Function[{v,w},Select[Tuples[{Range[Min@@Union[First/@prs],Max@@Union[First/@prs]],Range[Min@@Union[Last/@prs],Max@@Union[Last/@prs]]}],vdist[v,#]==1&&vdist[w,#]==1&]]@@@ovs]];
Table[Length[Select[Subsets[Tuples[Range[n],2]],flowdown[#]==#&]],{n,0,3}]

a(5)-a(6) from Christopher Cormier, Dec 10 2019

a(7)-a(15) from Christopher Cormier, Dec 19 2019

A344114 a(n) = 2^(n^2) - n!.

Original entry on oeis.org

1, 14, 506, 65512, 33554312, 68719476016, 562949953416272, 18446744073709511296, 2417851639229258349049472, 1267650600228229401496699576576, 2658455991569831745807614120520772352, 22300745198530623141535718272648361026978816, 748288838313422294120286634350736906063831234982912

Offset: 1

Mohammad K. Azarian, Jun 04 2021

a(n) is the number of relations on a set with n elements that are not one-to-one functions.

			a(1) = 2^(1^2) - 1! =   1;
a(2) = 2^(2^2) - 2! =  14;
a(3) = 2^(3^2) - 3! = 506.

Michael De Vlieger, Table of n, a(n) for n = 1..57
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A101030, A199656, A344110, A344112, A344113.

Mathematica
```
Table[2^(n^2) - n!, {n, 16}] // Flatten
```

A055548 Number of normal n X n (-1,1)-matrices.

Original entry on oeis.org

2, 12, 80, 2096, 49792, 3449088, 357236224, 84783217408

Offset: 1

Eric W. Weisstein

Obviously a(n) <= 2^(n^2) = A002416(n) - R. J. Mathar, Mar 14 2006

W. H. Press et al., Numerical Recipes, Cambridge, 1986; Chapter 11.

Georg Muntingh, Sage code for computing higher order entries
Eric Weisstein's World of Mathematics, Normal matrix.

Cf. A055547, A055549.

NormaQ(a,n) = { my(aT) ; aT=mattranspose(a) ; return( a*aT == aT*a ); }
combMat(no,n) = { my(a,noshif) ; a = matrix(n,n) ; noshif=no ; for(co=1,n, for(ro=1,n, if( (noshif %2)== 1,a[ro,co] = 1, a[ro,co] = -1) ; noshif = floor(noshif/2) ; ) ) ; return(a) ; }
{ for (n = 1, 10, count = 0; a = matrix(n,n) ; for( no=0,2^(n^2)-1, a = combMat(no,n) ; count += NormaQ(a,n) ; /* if(no%1000==0,print(n," ",(no/2^(n^2)+0.)," ",count)) ; */ ) ; print(count) ; ) } \\ R. J. Mathar, Mar 14 2006

a(5) from R. J. Mathar, Mar 14 2006
a(6)-a(7) from Georg Muntingh, Jan 31 2014
Offset corrected and a(8) from Bert Dobbelaere, Sep 21 2020

A086324 Number of n X n circulant singular matrices over GF(2).

Original entry on oeis.org

1, 2, 5, 8, 17, 40, 79, 128, 323, 544, 1025, 2560, 4097, 10112, 22643, 32768, 66047, 165376, 262145, 557056, 1513709, 2099200, 4198399, 10485760, 17825807, 33562624, 84672701, 165675008, 268435457, 741965824, 1259979967, 2147483648, 5378137091, 8656912384

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003473, A002416.

a(n) = 2^n - A003473(n).

a(10)-a(34) from Alois P. Heinz, Mar 16 2017

A135748 a(n) = Sum_{k=0..n} binomial(n,k)*2^(k^2).

Original entry on oeis.org

1, 3, 21, 567, 67689, 33887403, 68921796861, 563431696713567, 18451249599365935569, 2418017680197896730749523, 1267674779574792745831097365221, 2658469935859419140387217204140789127, 22300777100086187451068223319189800258419769

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

a(n) is the number of directed graphs on any subset of a set of n labeled nodes, allowing self-loops (cf. A002416). - Brent A. Yorgey, Mar 23 2021

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A002416.

Table[Sum[Binomial[n,k]2^k^2,{k,0,n}],{n,0,15}] (* Harvey P. Dale, May 30 2013 *)

PARI
```
{a(n)=sum(k=0,n,binomial(n,k)*2^(k^2))}
```

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Nov 27 2017

A158099 Euler transform of square powers of 2: [2,2^4,2^9,...,2^(n^2),...].

Original entry on oeis.org

1, 2, 19, 548, 66749, 33695574, 68787981855, 563088066184424, 18447871299903970005, 2417888543453357864445634, 1267655436282309648681395304255, 2658458526916981532120588021462151100, 22300750515466692968838881088968809185127601

Offset: 0

Paul D. Hanna, Mar 20 2009

nonn

			G.f.: A(x) = 1 + 2*x + 19*x^2 + 548*x^3 + 66749*x^4 +...
A(x) = 1/[(1-x)^2*(1-x^2)^(2^4)*(1-x^3)^(2^9)*(1-x^4)^(2^16)*...].

Alois P. Heinz, Table of n, a(n) for n = 0..35

Cf. A002416, A034899, A158098.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:= etr(n->2^(n^2)):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 03 2012

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Function[{n}, 2^(n^2)]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

a(n)=polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^(2^(k^2))),n)

{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d*2^(d^2))*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Oct 18 2009

G.f.: A(x) = 1/Product_{n>=1} (1 - x^n)^(2^(n^2)).

G.f.: exp( Sum_{n>=1} L(n)*x^n/n ) where L(n) = Sum_{d|n} d*2^(d^2). [Paul D. Hanna, Oct 18 2009]

A344115 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not one-to-one functions.

Original entry on oeis.org

1, 2, 14, 5, 58, 506, 12, 244, 4072, 65512, 27, 1004, 32708, 1048456, 33554312, 58, 4066, 262024, 16776856, 1073741104, 68719476016, 121, 16342, 2096942, 268434616, 34359735848, 4398046506064, 562949953416272, 248, 65480, 16776880, 4294965616, 1099511621056

Offset: 1

Mohammad K. Azarian, Jun 06 2021

If n=k, then T(n,k) = 2^(n^2) - n!, which is A344114, and if kA344110.

			For T(2,2): the number of relations is 2^4 and the number of one-to-one functions is 2, so 2^4 - 2 = 14 and thus T(2,2) = 14.
Triangle T(n,k) begins:
   1;
   2,   14;
   5,   58,   506;
  12,  244,  4072,   65512;
  27, 1004, 32708, 1048456, 33554312;

Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A068424, A101030, A199656, A344110, A344112, A344113, A344114.

Table[2^(n*k) - k!/(k - n)!, {k, 10}, {n, k}] // Flatten

T(n,k) = 2^(n*k) - k!/(k-n)!, k >= n.

cp's OEIS Frontend

A273461 Number of physically stable n X n placements of water source-blocks in Minecraft.

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A274805 The logarithmic transform of sigma(n).

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A283500 Triangle read by rows: T(n,k) = number of n X n (0,1) matrices with at most k 1's in each row or column.

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A329871 Number of static n X n placements of water source-blocks in Minecraft.

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Mathematica

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A344114 a(n) = 2^(n^2) - n!.

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Mathematica

A055548 Number of normal n X n (-1,1)-matrices.

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PARI

Extensions

A086324 Number of n X n circulant singular matrices over GF(2).

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A135748 a(n) = Sum_{k=0..n} binomial(n,k)*2^(k^2).

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