A144512 -id:A144512 - cp's OEIS frontend

A147984 Column 5 of A144512.

1, 2431, 4061871, 7528988476, 15467641899285, 34155922905682979, 79397199549271412737, 191739533381111401455478, 476872353039366288373555323

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144512, A048775, A144511, A144662.

A149187 a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., 6, for 0 <= k <= 6n.

Original entry on oeis.org

1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, 8064519699524417149584982475, 12261371699318896159811165091392898, 34949877647533654983311522321749656046802, 174047342897498341701547082125166096889157924610, 1431472607165249058159939223685478666695036430843693596

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most 5 times. - N. J. A. Sloane, Jan 25 2017

Alois P. Heinz, Table of n, a(n) for n = 0..100
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix I to "Analysis of the gift exchange problem", giving Type D recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix II to "Analysis of the gift exchange problem", giving Type C recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Cf. A144512.

The gift scenarios sequences when a gift can be stolen at most s times, for s = 1..9, are A001515, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.

with(combinat):
b:= proc(n, i, t) option remember; `if`(t*i add(b(k, 6, n), k=0..6*n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 17 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, t_] := b[n, i, t] = If[t*i < n, 0, If[n == 0, If[t == 0, 1, 0], Sum[b[n-i*j, i-1, t-j]* multinomial[n, Prepend[Array[i&, j], n-i*j]]/j!, {j, 0, Min[t, n/i]}]]]; a[n_] := Sum[b[k, 6, n], {k, 0, 6*n}];  Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)

{a(n) = sum(i=n, 6*n, i!*polcoef(sum(j=1, 6, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

A144510 Array T(n,k) (n >= 1, k >= 0) read by downwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., i] into exactly k nonempty blocks, each of size at most n, for any i in the range n <= i <= k*n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 37, 31, 4, 1, 1, 266, 842, 121, 5, 1, 1, 2431, 45296, 18252, 456, 6, 1, 1, 27007, 4061871, 7958726, 405408, 1709, 7, 1, 1, 353522, 546809243, 7528988476, 1495388159, 9268549, 6427, 8, 1

Offset: 1

David Applegate and N. J. A. Sloane, Dec 15 2008, Jan 30 2009

			Array begins:
1, 1,    1,       1,            1,                 1,                       1, ...
1, 2,    7,      37,          266,              2431,                   27007, ...
1, 3,   31,     842,        45296,           4061871,               546809243, ...
1, 4,  121,   18252,      7958726,        7528988476,          13130817809439, ...
1, 5,  456,  405408,   1495388159,    15467641899285,      361207016885536095, ...
1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...
...

Seiichi Manyama, Antidiagonals n = 1..50, flattened

For the transposed array see A144512.
Rows include A001515, A144416, A144508, A144509.
Columns include A048775, A144511.
A(n+1,n) gives A281901.
A(n,n) gives A308296.
Cf. A308292.

b := proc(n, i, k) local r;
option remember;
if n = i then 1;
elif i < n then 0;
elif n < 1 then 0;
else add( binomial(i-1,r)*b(n-1,i-1-r,k), r=0..k);
end if;
end proc;
T:=proc(n,k); add(b(n,i,k),i=0..(k+1)*n); end proc;
# Peter Luschny, Apr 26 2011
A144510 := proc(n, k) local m;
add(m!*coeff(expand((exp(x)*GAMMA(n+1,x)/GAMMA(n+1)-1)^k),x,m),m=k..k*n)/k! end: for row from 1 to 6 do seq(A144510(row, col), col = 0..5) od;

multinomial[n_, k_List] := n!/Times @@ (k!); t[n_, k_] := Module[{i, ik}, ik = Array[i, k]; 1/k!* Sum[multinomial[Total[ik], ik], Evaluate[Sequence @@ Thread[{ik, 1, n}]]]]; Table[t[n-k, k], {n, 1, 10}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

T(n,k) = (1/k!)*Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} multinomial(i_1+i_2+...+i_k; i_1, i_2, ..., i_k).

T(n,k) = (1/k!)*Sum_{m=k..k*n} m! [x^m](e^x Gamma(n+1,x)/Gamma(n+1)-1)^k. Here [x^m]f(x) is the coefficient of x^m in the series expansion of f(x). - Peter Luschny, Apr 26 2011

A144662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} (i+j+k+l)!/(4!i!j!k!l!).

Original entry on oeis.org

0, 1, 266, 45296, 7958726, 1495388159, 295887993624, 60790021361170, 12845435390707724, 2774049143394729653, 609542744597785306189, 135840016223787254538508, 30629983532857972983331740, 6975352854342057056747327899, 1602003695575764851150428242804, 370631496919828403109950449644134

Offset: 0

N. J. A. Sloane, Feb 01 2009

nonn

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Column 4 of A144512. Cf. A144660, A144661.

f:=n->add( add( add( add( (i+j+k+l)!/(4!*i!*j!*k!*l!), i=1..n),j=1..n),k=1..n),l=1..n); [seq(f(n),n=0..16)];

a[n_] := Sum[(i+j+k+l)!/(4! i! j! k! l!), {i, n}, {j, n}, {k, n}, {l, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 05 2018 *)
Table[(Binomial[2*n + 2, n + 1] - 2*(1 + n) + Sum[(1 + k + l + 2*n)! HypergeometricPFQ[{1, -1 - k - l - n, -n}, {-1 - k - l - 2*n, -k - l - n}, 1]/((1 + k + l + n) k! l! (n!)^2) - (2*(1 + k + l + n)!)/((1 + k + l) k! l! n!), {k, 1, n}, {l, 1, n}])/24, {n, 0, 15}] (* Vaclav Kotesovec, Apr 04 2019 *)

A281901 Number of scenarios in the Gift Exchange Game with n players and n wrapped gifts when a gift can be stolen at most n times.

Original entry on oeis.org

1, 2, 31, 18252, 1495388159, 34155922905682979, 350521520018942991464535019, 2371013832433361706367594400829713564440, 14584126149704606223764458141727351569547933381159988406, 107640669875812795238625627484701500354901860426640161278022882392148747562

Offset: 0

Alois P. Heinz, Feb 01 2017

nonn

Also total number of partitions of [k] into exactly n nonempty blocks, each of size at most n+1, for any k in the range n <= k <= n^2+n.

Alois P. Heinz, Table of n, a(n) for n = 0..26
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix I to "Analysis of the gift exchange problem"
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix II to "Analysis of the gift exchange problem"
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Main diagonal of A144510, A144512.

Cf. A001515, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.

with(combinat):
b:= proc(n, i, t) option remember; `if`(t*i add(b(j, n+1, n), j=0..(n+1)*n):
seq(a(n), n=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, t_] := b[n, i, t] = If[t*iJean-François Alcover, Mar 13 2017, translated from Maple *)

a(n) = A144510(n+1,n) = A144512(n,n).

A308296 a(n) = (1/n!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_n=1..n} multinomial(i_1+i_2+...+i_n; i_1, i_2, ... , i_n).

Original entry on oeis.org

1, 1, 7, 842, 7958726, 15467641899285, 10893033763705794846727, 4247805448772073978048752721163278, 1299618941291522676629215597535104557826094801396, 419715170056359079715862408734598208208707081189266290220651371206

Offset: 0

Seiichi Manyama, May 19 2019

nonn

			a(2) = (1/2) * (binomial(1+1,1) + binomial(1+2,2) + binomial(2+1,1) + binomial(2+2,2)) = 7.

Seiichi Manyama, Table of n, a(n) for n = 0..27

Cf. A144510, A144512, A274762, A281901.

{a(n) = sum(i=n, n^2, i!*polcoef(sum(j=1, n, x^j/j!)^n, i))/n!}

a(n) = A144510(n,n).

cp's OEIS Frontend

A147984 Column 5 of A144512.

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A149187 a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., 6, for 0 <= k <= 6n.

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A144510 Array T(n,k) (n >= 1, k >= 0) read by downwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., i] into exactly k nonempty blocks, each of size at most n, for any i in the range n <= i <= k*n.

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Programs

Maple

Mathematica

Formula

A144662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} (i+j+k+l)!/(4!*i!*j!*k!*l!).

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A281901 Number of scenarios in the Gift Exchange Game with n players and n wrapped gifts when a gift can be stolen at most n times.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A308296 a(n) = (1/n!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_n=1..n} multinomial(i_1+i_2+...+i_n; i_1, i_2, ... , i_n).

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Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A144662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} (i+j+k+l)!/(4!i!j!k!l!).