A144509 -id:A144509 - cp's OEIS frontend

A001515 Bessel polynomial y_n(x) evaluated at x=1.

1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106, 36496226977, 841146804577, 21065166341402, 569600638022431, 16539483668991901, 513293594376771362, 16955228098102446847, 593946277027962411007, 21992967478132711654106, 858319677924203716921141

Offset: 0

N. J. A. Sloane

For some applications it is better to start this sequence with an extra 1 at the beginning: 1, 1, 2, 37, 266, 2431, 27007, 353522, 5329837, ... (again with offset 0). This sequence now has its own entry - see A144301.
Number of partitions of {1,...,k}, n <= k <= 2n, into n blocks with no more than 2 elements per block. Restated, number of ways to use the elements of {1,...,k}, n <= k <= 2n, once each to form a collection of n sets, each having 1 or 2 elements. - Bob Proctor, Apr 18 2005, Jun 26 2006. E.g., for n=2 we get: (k=2): {1,2}; (k=3): {1,23}, {2,13}, {3,12}; (k=4): {12,34}, {13,24}, {14,23}, for a total of a(2) = 7 partitions.
Equivalently, number of sequences of n unlabeled items such that each item occurs just once or twice (cf. A105749). - David Applegate, Dec 08 2008
Numerator of (n+1)-th convergent to 1+tanh(1). - Benoit Cloitre, Dec 20 2002
The following Maple lines show how this sequence and A144505, A144498, A001514, A144513, A144506, A144514, A144507, A144301 are related.
f0:=proc(n) local k; add((n+k)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f0(n),n=0..10)];
# that is this sequence
f1:=proc(n) local k; add((n+k+1)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f1(n),n=0..10)];
# that is A144498
f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..10)];
# that is A144513; divided by 2 gives A001514
f3:=proc(n) local k; add((n+k+3)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f3(n),n=0..10)];
# that is A144514; divided by 6 gives A144506
f4:=proc(n) local k; add((n+k+4)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f4(n),n=0..10)];
# that divided by 24 gives A144507
a(n) is also the numerator of the continued fraction sequence beginning with 2 followed by 3 and the remaining odd numbers: [2,3,5,7,9,11,13,...]. - Gil Broussard, Oct 07 2009
Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most once. - N. J. A. Sloane, Jan 25 2017

			The first few Bessel polynomials are (cf. A001497, A001498):
  y_0 = 1
  y_1 = 1 +   x
  y_2 = 1 + 3*x +  3*x^2
  y_3 = 1 + 6*x + 15*x^2 + 15*x^3, etc.
G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 266*x^4 + 2431*x^5 + 27007*x^6 + 353522*x^7 + ...

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).

Seiichi Manyama, Table of n, a(n) for n = 0..404 (first 101 terms from T. D. Noe)
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.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials, arXiv:quant-ph/0501155, 2005.
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 11.
O. Frink and H. L. Krall, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65,100-115, 1945. [From _Roger L. Bagula_, Feb 15 2009]
E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
Wojciech Mlotkowski and Anna Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1968
J. Riordan, Notes 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
Index entries for related partition-counting sequences

Cf. A000108, A000806, A001514, A105749, A143990, A144505, A144506, A144507, A144513, A144514.
See A144301 for other formulas and comments.
Row sums of Bessel triangle A001497 as well as of A001498.
Partial sums: A105748.
First differences: A144498.
Replace "sets" with "lists" in comment: A001517.
The gift scenarios sequences when a gift can be stolen at most s times, for s = 1..9, are this sequence, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.

a001515 = sum . a001497_row -- Reinhard Zumkeller, Nov 24 2014

[(&+[Binomial(n+j, 2*j)*Catalan(j)*Factorial(j+1)/2^j: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Sep 26 2023

A001515 := proc(n) option remember; if n=0 then 1 elif n=1 then 2 else (2*n-1)*A001515(n-1)+A001515(n-2); fi; end;
A001515:=proc(n) local k; add( (n+k)!/((n-k)!*k!*2^k),k=0..n); end;
A001515:= n-> hypergeom( [n+1,-n],[],-1/2);
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==(2n-1)a[n-1]+a[n-2]},a[n], {n,25}] (* Harvey P. Dale, Jun 18 2011 *)
Table[Sum[BellY[n+1, k, (2 Range[n+1] - 3)!!], {k, n+1}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

{a(n) = if( n<0, n = -1 - n); sum( k=0, n, (2*n - k)! / (k! * (n-k)!) * 2^(k-n))} /* Michael Somos, Apr 08 2012 */

[sum(binomial(n+j,2*j)*binomial(2*j,j)*factorial(j)//2^j for j in range(n+1)) for n in range(31)] # G. C. Greubel, Sep 26 2023

The following formulas can all be found in (or are easily derived from formulas in) Grosswald's book.
D-finite with recurrence: a(0) = 1, a(1) = 2; thereafter a(n) = (2*n-1)*a(n-1) + a(n-2).
E.g.f.: exp(1-sqrt(1-2*x))/sqrt(1-2*x).
a(n) = Sum_{ k = 0..n } binomial(n+k,2*k)*(2*k)!/(k!*2^k).
Equivalently, a(n) = Sum_{ k = 0..n } (n+k)!/((n-k)!*k!*2^k) = Sum_{ k = n..2n } k!/((2n-k)!*(k-n)!*2^(k-n)).
a(n) = Hypergeometric2F0( [n+1, -n] ; - ; -1/2).
a(n) = A105749(n)/n!.
a(n) ~ exp(1)*(2n)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]
a(n) = A144301(n+1).
G.f.: 1/(1-x-x/(1-x-2*x/(1-x-3*x/(1-x-4*x/(1-x-5*x/(1-.... (continued fraction). - Paul Barry, Feb 08 2009
From Michael Somos, Apr 08 2012: (Start)
a(-1 - n) = a(n).
(a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. (End)
G.f.: 1/G(0) where G(k) = 1 - x - x*(2*k+1)/(1 - x - 2*x*(k+1)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2012
E.g.f.: E(0)/(2*sqrt(1-2*x)), where E(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)*(1+sqrt(1-2*x))/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013
G.f.: T(0)/(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
a(n) = (2*BesselI(1/2, 1)+BesselI(3/2, 1))*BesselK(n+1/2, 1). - Jean-François Alcover, Feb 03 2014
a(n) = exp(1)*sqrt(2/Pi)*BesselK(1/2+n,1). - Gerry Martens, Jul 22 2015
From Peter Bala, Apr 14 2017: (Start)
a(n) = (1/n!)*Integral_{x = 0..inf} exp(-x)*x^n*(1 + x/2)^n dx.
E.g.f.: d/dx( exp(x*c(x/2)) ) = 1 + 2*x + 7*x^2/2! + 37*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
From G. C. Greubel, Aug 16 2017: (Start)
a(n) = (1/2)_{n} * 2^n * hypergeometric1f1(-n; -2*n; 2).
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; 2*t/(1-t)^2). (End)

Extensively edited by N. J. A. Sloane, Dec 07 2008

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.

Original entry on oeis.org

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).

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

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

A281358 Number of scenarios in the Gift Exchange Game when a gift can be stolen at most 6 times.

Original entry on oeis.org

1, 7, 6427, 216864652, 60790021361170, 79397199549271412737, 350521520018942991464535019, 4247805448772073978048752721163278, 122022975450467092259059357046375920848764, 7449370563518425038119522091529589590475534631830

Offset: 0

N. J. A. Sloane, Jan 25 2017

nonn

The result from the recurrence has been confirmed up to a(63) by using an optimized version of equation (23) in the Applegate-Sloane paper. - Lars Blomberg, Feb 01 2017

Lars Blomberg, Table of n, a(n) for n = 0..88
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.

with(combinat):
b:= proc(n, i, t) option remember; `if`(t*i add(b(k, 7, n), k=0..7*n):
seq(a(n), n=0..12);  # Alois P. Heinz, Feb 01 2017

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 7*n := t[n, k] = Sum[(1/j!)*Product[k - m, {m, 1, j}]*t[n - 1, k - j - 1], {j, 0, 6}]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 7*n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 7*n, i!*polcoef(sum(j=1, 7, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

More terms from Lars Blomberg, Feb 01 2017

A281359 Number of scenarios in the Gift Exchange Game when a gift can be stolen at most 7 times.

Original entry on oeis.org

1, 8, 24301, 5165454442, 12845435390707724, 191739533381111401455478, 11834912423104188943497126664597, 2371013832433361706367594400829713564440, 1299618941291522676629215597535104557826094801396, 1716119248126070756229849154290399886241087778087554633612

Offset: 0

N. J. A. Sloane, Jan 25 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..77
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.

with(combinat):
b:= proc(n, i, t) option remember; `if`(t*i add(b(k, 8, n), k=0..8*n):
seq(a(n), n=0..12);  # Alois P. Heinz, Feb 01 2017

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 8*n := t[n, k] = Sum[(1/j!)*Product[k - m, {m, 1, j}]*t[n - 1, k - j - 1], {j, 0, 7}]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 8*n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 8*n, i!*polcoef(sum(j=1, 8, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

A281360 Number of scenarios in the Gift Exchange Game when a gift can be stolen at most 8 times.

Original entry on oeis.org

1, 9, 92368, 124762262630, 2774049143394729653, 476872353039366288373555323, 414678423576860263798348331987688320, 1383884737648788823775562903922773021277571568, 14584126149704606223764458141727351569547933381159988406, 419715170056359079715862408734598208208707081189266290220651371206

Offset: 0

N. J. A. Sloane, Jan 25 2017

nonn

More than the usual number of terms are shown in the DATA field because there are the initial values needed for one of the recurrences.

Seiichi Manyama, Table of n, a(n) for n = 0..69
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.

with(combinat):
b:= proc(n, i, t) option remember; `if`(t*i add(b(k, 9, n), k=0..9*n):
seq(a(n), n=0..12);  # Alois P. Heinz, Feb 01 2017

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 9*n := t[n, k] = Sum[(1/j!)*Product[k - m, {m, 1, j}]*t[n - 1, k - j - 1], {j, 0, 8}]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 9*n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 9*n, i!*polcoef(sum(j=1, 9, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

A281361 Number of scenarios in the Gift Exchange Game when a gift can be stolen at most 9 times.

Original entry on oeis.org

1, 10, 352705, 3047235458767, 609542744597785306189, 1214103036523322674154687139158, 14963835327495031822418126706099787884130, 836883118002221273912672040462907783367741190535388, 170589804359366329173961838612841486616626580885839826818966688, 107640669875812795238625627484701500354901860426640161278022882392148747562, 185260259482556646382994900799988470488841686941141661692183483756531004879305895810561

Offset: 0

N. J. A. Sloane, Jan 25 2017

nonn

More than the usual number of terms are shown in the DATA field because there are the initial values needed for one of the recurrences.

Seiichi Manyama, Table of n, a(n) for n = 0..63
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.

with(combinat):
b:= proc(n, i, t) option remember; `if`(t*i add(b(k, 10, n), k=0..10*n):
seq(a(n), n=0..12);  # Alois P. Heinz, Feb 01 2017

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 10*n := t[n, k] = Sum[(1/j!)*Product[k - m, {m, 1, j}]*t[n - 1, k - j - 1], {j, 0, 9}]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 10*n}]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 10*n, i!*polcoef(sum(j=1, 10, 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

A144512 Array read by upwards antidiagonals: T(n,k) = total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2, ..., k+1, for 0 <= k <= (k+1)*n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 31, 37, 1, 1, 5, 121, 842, 266, 1, 1, 6, 456, 18252, 45296, 2431, 1, 1, 7, 1709, 405408, 7958726, 4061871, 27007, 1, 1, 8, 6427, 9268549, 1495388159, 7528988476, 546809243, 353522, 1, 1, 9, 24301, 216864652, 295887993624, 15467641899285

Offset: 0

David Applegate and N. J. A. Sloane, Dec 15 2008, Dec 21 2008

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, ...
1, 3, 31, 842, 45296, 4061871, 546809243, 103123135501, ...
1, 4, 121, 18252, 7958726, 7528988476, 13130817809439, ...
1, 5, 456, 405408, 1495388159, 15467641899285, 361207016885536095, ...
1, 6, 1709, 9268549, 295887993624, 34155922905682979, 10893033763705794846727, ...
...

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)

See A144510 for Maple code.
Rows include A001515, A144416, A144508, A144509, A149187.
Columns include A048775, A144511, A144662, A147984.
Transpose of array in A144510.
Main diagonal gives A281901.

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;

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, 0, n-1}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

cp's OEIS Frontend

A001515 Bessel polynomial y_n(x) evaluated at x=1.

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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.

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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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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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A281358 Number of scenarios in the Gift Exchange Game when a gift can be stolen at most 6 times.

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A281359 Number of scenarios in the Gift Exchange Game when a gift can be stolen at most 7 times.

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A281360 Number of scenarios in the Gift Exchange Game when a gift can be stolen at most 8 times.

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