A001515 -id:A001515 - cp's OEIS frontend

A144416 a(n) is the total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2 or 3, for 0 <= k <= 3n.

1, 3, 31, 842, 45296, 4061871, 546809243, 103123135501, 25942945219747, 8394104851717686, 3395846808758759686, 1679398297627675722593, 996789456118195908366641, 699283226713639676370419067, 572385833490097906671186099971, 540635257271794961275858251107746, 583630397618757664934692641037584628

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008, Dec 17 2008

nonn

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

			a(0) = 1;
a(1) = 3: {1} {12} {123}
a(2) = 31: {1,2} {1,23} {2,13} {3,12} {1,234} {2,134} {3,124} {4,123}
{12,34} {13,24} {14,23} {12,345} {13,245} {14,235} {15,234} {23,145} {24,135}
{25,134} {34,125} {35,124} {45,123} {123,456} {124,356} {125,346} {126,345}
{134,256} {135,246} {136,245} {145,236} {146,235} {156,234}.

David Applegate and N. J. A. Sloane, 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.

Row sums of A144385. Slice sums of A144626.

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.

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 3*n := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 3*n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 3*n, i!*polcoef(sum(j=1, 3, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

a(n) = Sum_{ b,c >= 0, b+c <= n } (n+b+2c)!/ ((n-b-c)! b! c! 2^b 6^c).

The sum is dominated by the b=0, c=n term, so a(n) ~ constant*(3*n)!/(n!*6^n).

A001514 Bessel polynomial {y_n}'(1).

Original entry on oeis.org

0, 1, 9, 81, 835, 9990, 137466, 2148139, 37662381, 733015845, 15693217705, 366695853876, 9289111077324, 253623142901401, 7425873460633005, 232122372003909045, 7715943399320562331, 271796943164015920914, 10114041937573463433966

Offset: 0

N. J. A. Sloane

nonn

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
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).

G. C. Greubel, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A001516, A001518, A065920, A144505.

(As in A001497 define:) f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
[seq( subs(x=1,diff(f(n),x)),n=0..60)];
f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..60)]; # uses a different offset

Table[Sum[(n+k+1)!/((n-k-1)!*k!*2^(k+1)), {k,0,n-1}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0}, Table[n*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[1 - n, -2*n, 2], {n,1,50}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,50, print1(sum(k=0,n-1, (n+k+1)!/((n-k-1)!*k!*2^(k+1))), ", ")) \\ G. C. Greubel, Aug 14 2017

a(n) = (1/2) * Sum_{k=0..n} (n+k+2)!/((n-k)!*k!*2^k) (with a different offset).
D-finite with recurrence: (n-1)^2 * a(n) = (2*n-1)*(n^2 - n + 1)*a(n-1) + n^2*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ 2^(n+1/2) * n^(n+1) / exp(n-1). - Vaclav Kotesovec, Jul 22 2015
a(n) = n*2^n*(1/2){n}*hypergeometric1f1(1-n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 14 2017
From G. C. Greubel, Aug 16 2017: (Start)
G.f.: (1/(1-t))*hypergeometric2f0(2, 3/2; -; 2*t/(1-t)^2).
E.g.f.: (1 - 2*x)^(-3/2)*((1 - x)*sqrt(1 - 2*x) + (3*x - 1))*exp((1 - sqrt(1 - 2*x))). (End)

A368598 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n}, or unlabeled loop-graphs with n edges and up to n vertices.

Original entry on oeis.org

1, 1, 2, 6, 17, 52, 173, 585, 2064, 7520, 28265, 109501, 437394, 1799843, 7629463, 33302834, 149633151, 691702799, 3287804961, 16058229900, 80533510224, 414384339438, 2185878202630, 11811050484851, 65318772618624, 369428031895444, 2135166786135671, 12601624505404858

Offset: 0

Gus Wiseman, Jan 05 2024

nonn

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}        {{1},{2},{3},{4}}
             {{1},{1,2}}  {{1},{2},{1,2}}      {{1},{2},{3},{1,2}}
                          {{1},{2},{1,3}}      {{1},{2},{3},{1,4}}
                          {{1},{1,2},{1,3}}    {{1},{2},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}    {{1},{2},{1,2},{3,4}}
                          {{1,2},{1,3},{2,3}}  {{1},{2},{1,3},{1,4}}
                                               {{1},{2},{1,3},{2,3}}
                                               {{1},{2},{1,3},{2,4}}
                                               {{1},{3},{1,2},{2,4}}
                                               {{1},{1,2},{1,3},{1,4}}
                                               {{1},{1,2},{1,3},{2,3}}
                                               {{1},{1,2},{1,3},{2,4}}
                                               {{1},{1,2},{2,3},{3,4}}
                                               {{2},{1,2},{1,3},{1,4}}
                                               {{4},{1,2},{1,3},{2,3}}
                                               {{1,2},{1,3},{1,4},{2,3}}
                                               {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Loop.

For any number of edges of any size we have A000612, covering A055621.
For any number of edges we have A000666, A054921, A322700.
The labeled version is A014068.
Counting by weight gives A320663, or A339888 with loops {x,x}.
The covering case is A368599.
For edges of any size we have A368731, covering A368186.
Row sums of A368836.
A000085 counts set partitions into singletons or pairs.
A001515 counts length-n set partitions into singletons or pairs.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A001434, A007716, A007717, A058891, A070166, A122848, A124059, A283877, A302545, A322661, A339741, A339887, A370168.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,2}],{n}]]],{n,0,5}]

a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A070166. - Andrew Howroyd, Jan 09 2024

a(n) = A070166(n, n). - Andrew Howroyd, Jan 09 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 09 2024

A144508 a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, 3 or 4, for 0 <= k <= 4n.

Original entry on oeis.org

1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, 38001495237579931, 169490425291053577442, 1102725620990181693266071, 10030550674270068548738783696, 123317200510025161580777179001154, 1993320784474917266370637900936051186, 41401645296339316791633672053851083955147

Offset: 0

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

nonn

Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most 3 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.

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.

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 4*n := t[n, k] = t[n - 1, k - 1] + (k - 1)*t[n - 1, k - 2] + (1/2)*(k - 1)*(k - 2)*t[n - 1, k - 3] + (1/6)*(k - 1)*(k - 2)*(k - 3)*t[n - 1, k - 4]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 4*n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 4*n, i!*polcoef(sum(j=1, 4, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

A368599 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n} with union {1..n}, or unlabeled loop-graphs with n edges covering n vertices.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 97, 277, 825, 2486, 7643, 23772, 74989, 238933, 769488, 2500758, 8199828, 27106647, 90316944, 303182461, 1025139840, 3490606305, 11967066094, 41302863014, 143493606215, 501772078429, 1765928732426, 6254738346969, 22294413256484, 79968425399831

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

			The a(0) = 1 through a(4) = 13 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}        {{1},{2},{3},{4}}
             {{1},{1,2}}  {{1},{2},{1,3}}      {{1},{2},{3},{1,4}}
                          {{1},{1,2},{1,3}}    {{1},{2},{1,2},{3,4}}
                          {{1},{1,2},{2,3}}    {{1},{2},{1,3},{1,4}}
                          {{1,2},{1,3},{2,3}}  {{1},{2},{1,3},{2,4}}
                                               {{1},{2},{1,3},{3,4}}
                                               {{1},{1,2},{1,3},{1,4}}
                                               {{1},{1,2},{1,3},{2,4}}
                                               {{1},{1,2},{2,3},{2,4}}
                                               {{1},{1,2},{2,3},{3,4}}
                                               {{1},{2,3},{2,4},{3,4}}
                                               {{1,2},{1,3},{1,4},{2,3}}
                                               {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Loop.

For any number of edges we have A000666, A054921, A322700.
For any number of edges of any size we have A055621, non-covering A000612.
For edges of any size we have A368186, covering case of A368731.
The labeled version is A368597, covering case of A014068.
This is the covering case of A368598.
A000085 counts set partitions into singletons or pairs.
A001515 counts length-n set partitions into singletons or pairs.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A007716, A007717, A058891, A070166, A122848, A283877, A302545, A320663, A322661, A339741, A339887, A339888.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}], Union@@#==Range[n]&]]],{n,0,5}]

a(n) = polcoef(G(n, O(x*x^n)) - if(n, G(n-1, O(x*x^n))), n) \\ G defined in A070166. - Andrew Howroyd, Jan 09 2024

a(n) = A070166(n,n) - A070166(n-1,n) for n > 0. - Andrew Howroyd, Jan 09 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 09 2024

A144509 a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., 5, for 0 <= k <= 5n.

Original entry on oeis.org

1, 5, 456, 405408, 1495388159, 15467641899285, 361207016885536095, 16557834064546698285496, 1350410785161120363519024709, 182141025850694258874753732988078, 38395944834298393218465758049745918098, 12093097322244029427838390643054170162054258, 5485321312184901565806045962453632525844948020084

Offset: 0

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

nonn

Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most 4 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.

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.

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 5*n := t[n, k] = Sum[(1/j!)*Product[k - m, {m, 1, j}]*t[n - 1, k - j - 1], {j, 0, 4}]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 5*n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 5*n, i!*polcoef(sum(j=1, 5, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

A001516 Bessel polynomial {y_n}''(1).

Original entry on oeis.org

0, 0, 6, 120, 1980, 32970, 584430, 11204676, 233098740, 5254404210, 127921380840, 3350718545460, 94062457204716, 2819367702529560, 89912640142178490, 3040986592542420060, 108752084073199561140, 4101112025363285051526

Offset: 0

N. J. A. Sloane

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
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).

G. C. Greubel, Table of n, a(n) for n = 0..400
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
Index entries for sequences related to Bessel functions or polynomials

Cf. A001497, A001498, A001514, A001515, A001518, A065944, A144505.

(As in A001497 define:) f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
[seq( subs(x=1,diff(f(n),x$2)),n=0..60)];

Table[Sum[(n+k+2)!/(2^(k+2)*(n-k-2)!*k!), {k,0,n-2}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0, 0}, Table[n*(n - 1)*Pochhammer[1/2, n]*2^n* Hypergeometric1F1[2 - n, -2*n, 2], {n,2,50}]] (* G. C. Greubel, Aug 14 2017 *)

for(n=0,20, print1(sum(k=0,n-2, (n+k+2)!/(2^(k+2)*(n-k-2)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017

G.f.: 6*x^2*(1-x)^(-5)*hypergeom([5/2,3],[],2*x/(x-1)^2). - Mark van Hoeij, Nov 07 2011
D-finite with recurrence: (n-2)*(n-1)*a(n) = (2*n - 1)*(n^2 - n + 2)*a(n-1) + n*(n+1)*a(n-2). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ 2^(n+1/2) * n^(n+2) / exp(n-1). - Vaclav Kotesovec, Jul 22 2015
a(n) = n*(n - 1)*(1/2){n}*2^n* hypergeometric1F1(2 - n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 14 2017
E.g.f.: (-1)*(1 - 2*x)^(-5/2)*((4 - 14*x + 9*x^2)*sqrt(1 - 2*x) + (2*x^3 - 24*x^2 + 18*x - 4))*exp((1 - sqrt(1 - 2*x))). - G. C. Greubel, Aug 16 2017

A144498 Column 2 of array in A144502.

Original entry on oeis.org

1, 5, 30, 229, 2165, 24576, 326515, 4976315, 85630914, 1642623355, 34762642871, 804650577600, 20224019536825, 548535471681029, 15969883030969470, 496754110707779461, 16441934503725675485, 576991048929859964160, 21399021201104749243099, 836326710446071005267035

Offset: 0

David Applegate and N. J. A. Sloane, Dec 13 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

First differences of A001515 and A144301.

A144498:= func< n | (&+[Binomial(n,k)*Factorial(n+k+1)/(2^k*Factorial(n)): k in [0..n]]) >;
[A144498(n): n in [0..30]]; // G. C. Greubel, Oct 07 2023

f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),n=0..60)];

Table[Sum[(n+k+1)!/((n-k)!*k!*2^k), {k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2013 *)

def A144498(n): return sum(binomial(n,k)*rising_factorial(n+1,k+1)//2^k for k in range(n+1))
[A144498(n) for n in range(31)] # G. C. Greubel, Oct 07 2023

a(n) = A001515(n+1) - A001515(n).
a(n) = A144301(n+2) - A144301(n+1).
E.g.f.: (1 - 2*x + 2*x*sqrt(1-2*x))*exp(1-sqrt(1-2*x))/(1-2*x)^2. - Sergei N. Gladkovskii, Oct 06 2012
G.f.: (1-x)/(x*Q(0)) - 1/x, where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n-1). - Vaclav Kotesovec, Oct 08 2013
G.f.: T(0)/x- 1/x, where T(k) = 1 - (k+1)*x/((k+1)*x - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2013
(2*n-1)*a(n) = (4*n^2 + 1)*a(n-1) + (2*n+1)*a(n-2). - G. C. Greubel, Oct 07 2023

A336400 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,2,1,...,1,2); a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 9, 44, 303, 2697, 29438, 380529, 5683359, 96290588, 1824544857, 38229811083, 877643031554, 21906313145979, 590665804363833, 17109084307014332, 529833078045763263, 17468521692479218209, 610901505126064857854, 22586913755160674065113

Offset: 0

Dmitry Efimov, Jul 22 2020

nonn

Number of perfect matchings of a chord diagram with 2*n vertices, where neighboring vertices are joined by two chords, and any other pair of vertices is joined by one chord.

			A symmetric 4x4 Toeplitz matrix A with the first row (0,2,1,2) has the form:
0 2 1 2
2 0 2 1
1 2 0 2
2 1 2 0.
Its hafnian equals Hf(A)=a12*a34+a13*a24+a14*a23=2*2+1*1+2*2=9.

Georg Fischer, Table of n, a(n) for n = 0..200
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2020.

Cf. A001515, A336114, A336286.

I:=[1,2,9]; [n le 3 select I[n] else ( (22*n^2-118*n+139)*Self(n-1) + (10*n^2-74*n+141)*Self(n-2) + 5*(n-5)*Self(n-3) )/(11*n-40): n in [1..30]]; // G. C. Greubel, Sep 28 2023

a:= proc(n) option remember; `if`(n<3, [1, 2, 9][n+1],
     ((22*n^2-74*n+43)*a(n-1)+(10*n^2-54*n+77)*a(n-2)
      +5*(n-4)*a(n-3))/(11*n-29))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 28 2020

Join[{1,2},Table[2^(n+1)*HypergeometricU[n,1+2 n,2],{n,2,19}]] (* Stefano Spezia, Jul 22 2020 *)
(* or *) Join[{1, 2}, Table[n*Sum[(n+k-1)!/(k!*(n-k)!*2^(k-1)),{k,0,n}],{n,2,200}]] (* Georg Fischer, Jul 28 2020 *)

def A336400(n): return n+1 if n<2 else (2/factorial(n-1))*sum(binomial(n,k)*gamma(n+k)/2^k for k in range(n+1))
[A336400(n) for n in range(31)] # G. C. Greubel, Sep 28 2023

a(n) = n*Sum_{k=0..n} (n+k-1)!/(k!*(n-k)!*2^(k-1)), n>=2.
a(n) = A001515(n) + A001515(n-1), n>=2.
a(n) ~ (2*n)!*e/(n!*2^n).
a(n) = 2^(n+1)*U(n, 1+2*n, 2) for n >= 2, where U(a, b, c) is the confluent hypergeometric function. - Stefano Spezia, Jul 22 2020
a(n) = ((22*n^2-74*n+43)*a(n-1) + (10*n^2-54*n+77)*a(n-2) + 5*(n-4)*a(n-3)) / (11*n-29) for n >= 3. - Alois P. Heinz, Jul 28 2020
E.g.f.: - 1 - x + (1 + 1/sqrt(1-2*x))*exp(1 - sqrt(1-2* x)). - G. C. Greubel, Sep 28 2023

Incorrect recurrences removed and a(18) corrected by Georg Fischer, Jul 28 2020

A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 3, 0, 0, 1, 10, 15, 0, 0, 0, 1, 15, 45, 15, 0, 0, 0, 1, 21, 105, 105, 0, 0, 0, 0, 1, 28, 210, 420, 105, 0, 0, 0, 0, 1, 36, 378, 1260, 945, 0, 0, 0, 0, 0, 1, 45, 630, 3150, 4725, 945, 0, 0, 0, 0, 0, 1, 55, 990, 6930, 17325, 10395, 0, 0, 0, 0, 0, 0

Offset: 0

David Applegate and N. J. A. Sloane, Dec 06 2008

T(n,k) is the number of partitions of an n-set into k nonempty subsets, each of size at most 2.
The Grosswald and Choi-Smith references give many further properties and formulas.
Considered as an infinite lower triangular matrix T, lim_{n->infinity} T^n = A118930: (1, 1, 2, 4, 13, 41, 166, 652, ...) as a vector. - Gary W. Adamson, Dec 08 2008

			Triangle begins:
  n:
  0: 1
  1: 1  0
  2: 1  1   0
  3: 1  3   0    0
  4: 1  6   3    0   0
  5: 1 10  15    0   0  0
  6: 1 15  45   15   0  0  0
  7: 1 21 105  105   0  0  0  0
  8: 1 28 210  420 105  0  0  0  0
  9: 1 36 378 1260 945  0  0  0  0  0
  ...
The row sums give A000085.
For some purposes it is convenient to rotate the triangle by 45 degrees:
  1 0 0 0 0  0  0   0   0    0    0     0 ...
    1 1 0 0  0  0   0   0    0    0     0 ...
      1 3 3  0  0   0   0    0    0     0 ...
        1 6 15 15   0   0    0    0     0 ...
          1 10 45 105 105    0    0     0 ...
             1 15 105 420  945  945     0 ...
                1  21 210 1260 4725 10395 ...
                    1  28  378 3150 17325 ...
                        1   36  630  6930 ...
                             1   45   990 ...
  ...
The latter triangle is important enough that it has its own entry, A144331. Here the column sums give A000085 and the rows sums give A001515.
If the entries in the rotated triangle are denoted by b1(n,k), n >= 0, k <= 2n, then we have the recurrence b1(n, k) = b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2).
Then b1(n,k) is the number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1 or 2.

E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
Toufik Mansour, Matthias Schork, and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
T. Mansour and M. Shattuck, Partial matchings and pattern avoidance, Appl. Anal. Discrete Math. 7 (2013) 25-50.

Other versions of this same triangle are given in A111924 (which omits the first row), A001498 (which left-adjusts the rows in the bottom view), A001497 and A100861. Row sums give A000085.

Cf. A000085, A000217, A001464, A004526, A118930, A144385, A144643..

a144299 n k = a144299_tabl !! n !! k
a144299_row n = a144299_tabl !! n
a144299_tabl = [1] : [1, 0] : f 1 [1] [1, 0] where
   f i us vs = ws : f (i + 1) vs ws where
               ws = (zipWith (+) (0 : map (i *) us) vs) ++ [0]
-- Reinhard Zumkeller, Jan 01 2014

A144299:= func< n,k | k le Floor(n/2) select Factorial(n)/(Factorial(n-2*k)*Factorial(k)*2^k) else 0 >;
[A144299(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2023

Maple code producing the rotated version:
b1 := proc(n, k)
option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2);
end if;
end proc;
for n from 0 to 12 do lprint([seq(b1(n,k),k=0..2*n)]); od:

T[n_,0]=0; T[1,1]=1; T[2,1]=1; T[n_, k_]:= T[n-1,k-1] + (n-1)T[n-2,k-1];
Table[T[n,k], {n,12}, {k,n,1,-1}]//Flatten (* Robert G. Wilson v *)
Table[If[k<=Floor[n/2],n!/((n-2 k)! k! 2^k),0], {n, 0, 12},{k,0,n}]//Flatten (* Stefano Spezia, Jun 15 2023 *)

def A144299(n,k): return factorial(n)/(factorial(n-2*k)*factorial(k)*2^k) if k <= (n//2) else 0
flatten([[A144299(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 29 2023

T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).
E.g.f.: Sum_{k >= 0} Sum_{n = 0..2k} T(n,k) y^k x^n/n! = exp(y(x+x^2/2)). (The coefficient of y^k is the e.g.f. for the k-th row of the rotated triangle shown below.)
T(n, k) = n!/((n - 2*k)!*k!*2^k) for 0 <= k <= floor(n/2) and 0 otherwise. - Stefano Spezia, Jun 15 2023
From G. C. Greubel, Sep 29 2023: (Start)
T(n, 1) = A000217(n-1).
Sum_{k=0..n} T(n,k) = A000085(n).
Sum_{k=0..n} (-1)^k*T(n,k) = A001464(n). (End)

Offset fixed by Reinhard Zumkeller, Jan 01 2014

cp's OEIS Frontend

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A368598 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n}, or unlabeled loop-graphs with n edges and up to n vertices.

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A144508 a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, 3 or 4, for 0 <= k <= 4n.

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A368599 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n} with union {1..n}, or unlabeled loop-graphs with n edges covering n vertices.

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A144509 a(n) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., 5, for 0 <= k <= 5n.

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A001516 Bessel polynomial {y_n}''(1).

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