A001515 -id:A001515 - cp's OEIS frontend

A105748 Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a collection of n (possibly empty) sets, each with at most 2 elements.

1, 3, 10, 47, 313, 2744, 29751, 383273, 5713110, 96673861, 1830257967, 38326484944, 879473289521, 21944639630923, 591545277653354, 17131028946645255, 530424623323416617, 17485652721425863464, 611431929749388274471, 22604399407882099928577

Offset: 0

Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005

			a(2) = 10 = |{ {{},{}}, {{},{1}}, {{},{1,2}}, {{1},{2}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}} }|.

Alois P. Heinz, Table of n, a(n) for n = 0..400
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math.CO.0606404.
Index entries for related partition-counting sequences

First differences: A001515.
Replacing "collection" by "sequence" gives A003011.
Replacing "sets" by "lists" gives A105747.

a:= proc(n) option remember; `if`(n<3, [1, 3, 10][n+1],
      2*n*a(n-1)-(2*n-2)*a(n-2)-a(n-3))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 11 2015

Sum[(k+i)!/i!/(k-i)!/2^i, {k, 0, n}, {i, 0, k}]
(* Second program: *)
a[n_] := E*Sqrt[2/Pi]*Sum[BesselK[k + 1/2, 1], {k, 0, n}]; Table[a[n] // Round, {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017 *)

A105748(n) = sum(k=0,n,sum(i=0,k, binomial(k+i,k-i)*binomial(2*i,i)*i!>>i))  \\ M. F. Hasler, Oct 09 2012

a(n) = Sum_{0<=i<=k<=n} (k+i)!/i!/(k-i)!/2^i.
G.f.: 1/U(0)  where U(k)= 1 - 3*x + x^2 - x*4*k - x^2*(2*k+1)*(2*k+2)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 06 2012
G.f.: 1/(1-x)/Q(0), where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
a(n) = 2*n*a(n-1) -(2*n-2)*a(n-2) -a(n-3) for n>2. - Alois P. Heinz, Mar 11 2015
a(n) ~ 2^(n + 1/2) * n^n / exp(n-1). - Vaclav Kotesovec, May 05 2024

A107102 Matrix inverse of A100862.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -7, 12, -6, 1, 37, -67, 39, -10, 1, -266, 495, -310, 95, -15, 1, 2431, -4596, 3000, -1010, 195, -21, 1, -27007, 51583, -34566, 12320, -2660, 357, -28, 1, 353522, -680037, 463981, -171766, 39795, -6062, 602, -36, 1, -5329837, 10306152, -7124454, 2709525, -658791, 108927, -12432, 954

Offset: 0

Paul D. Hanna, May 21 2005

Column 0 is signed A001515 (Bessel polynomial). Column 1 is A107103. Row sums are zeros for n>0. Absolute row sums form A107104, which equals 2*A043301(n-1) for n>0.

The row polynomials p_n(x) of this entry are (-1)^n B_n(1-x), where B_n(x) are the modified Carlitz-Bessel polynomials of A001497, e,g, (-1)^2 B_2(1-x) = (1-x) + (1-x)^2 = 2 - 3 x + x^2 = p_2(x). - Tom Copeland, Oct 10 2016

			Triangle begins:
1;
-1,1;
2,-3,1;
-7,12,-6,1;
37,-67,39,-10,1;
-266,495,-310,95,-15,1;
2431,-4596,3000,-1010,195,-21,1;
-27007,51583,-34566,12320,-2660,357,-28,1; ...
and is the matrix inverse of A100862:
1;
1,1;
1,3,1;
1,6,6,1;
1,10,21,10,1;
1,15,55,55,15,1; ...

Cf. A100862, A001515, A043301, A107103, A107104.

Cf. A001497.

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^n));(matrix(n+1,n+1,m,j,if(m>=j, (m-1)!*polcoeff(polcoeff(exp(X+Y*X^2/2+X*Y),m-1,x),j-1,y)))^-1)[n+1,k+1]}

E.g.f.: exp((1-y)*(1-sqrt(1+2*x))). [Vladeta Jovovic, Dec 13 2008]

A143968 Denominators of numbers with g.f. exp(1-(1-x)^(1/2)).

Original entry on oeis.org

1, 2, 4, 48, 384, 1920, 46080, 645120, 5160960, 185794560, 530841600, 40874803200, 280284364800, 51011754393600, 714164561510400, 42849873690624000, 1371195958099968000, 302731575164928000, 1678343852714360832000, 100120983364435968000, 1275541328062914232320000

Offset: 0

N. J. A. Sloane, Dec 02 2008

			1, 1/2, 1/4, 7/48, 37/384, 133/1920, 2431/46080, 27007/645120, 176761/5160960, ...

Robert Israel, Table of n, a(n) for n = 0..406

Cf. A143991 (numerators), A001515.

S:= series(exp(1-(1-x)^(1/2)),x,21):
seq(denom(coeff(S,x,i)),i=0..20); # Robert Israel, Mar 23 2023

CoefficientList[Series[Exp[1-Sqrt[1-x]],{x,0,20}],x]//Denominator (* Harvey P. Dale, Sep 13 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).

A144513 a(n) = Sum_{k=0..n} (n+k+2)!/((n-k)!k!2^k).

Original entry on oeis.org

2, 18, 162, 1670, 19980, 274932, 4296278, 75324762, 1466031690, 31386435410, 733391707752, 18578222154648, 507246285802802, 14851746921266010, 464244744007818090, 15431886798641124662, 543593886328031841828, 20228083875146926867932, 792934721766833544369830

Offset: 0

N. J. A. Sloane, Dec 16 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..402

Equals 2*A001514 (with a different offset).

Cf. A001515, A144498, A144514.

f2:=proc(n) local k; add((n+k+2)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f2(n),n=0..50)];

{a(n) = sum(k=0, n, (n+k+2)!/((n-k)!*k!*2^k))} \\ Seiichi Manyama, Apr 07 2019

n^2*a(n) = (2*n+1)*(n^2+n+1)*a(n-1) + (n+1)^2*a(n-2). - Seiichi Manyama, Apr 07 2019

A144514 a(n) = Sum_{k=0..n} (n+k+3)!/((n-k)!k!2^k).

Original entry on oeis.org

6, 84, 1050, 13980, 205800, 3368316, 61075854, 1219445100, 26635157010, 632479986600, 16235529291696, 448220024574504, 13247429692101150, 417453231024613140, 13974133833217747650, 495278130521939366196, 18530507890959175097784, 729908595489477119015700

Offset: 0

N. J. A. Sloane, Dec 16 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..401

Equals 6*A144506 (with a different offset).

Cf. A001515, A144498, A144513.

f3:=proc(n) local k; add((n+k+3)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f3(n),n=0..50)];

Table[Sum[(n+k+3)!/((n-k)!k! 2^k),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 27 2019 *)

{a(n) = sum(k=0, n, (n+k+3)!/((n-k)!*k!*2^k))} \\ Seiichi Manyama, Apr 07 2019

a(n) ~ 2^(n + 7/2) * n^(n+3) / exp(n-1). - Vaclav Kotesovec, Apr 07 2019

A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 15, 15, 0, 1, 10, 45, 105, 105, 0, 1, 15, 105, 420, 945, 945, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 1, 36, 630, 6930, 51975, 270270, 945945, 2027025, 2027025, 0

Offset: 0

Eric W. Weisstein, Mar 14 2005

			Bessel polynomials begin with:
      x;
      x +     x^2;
    3*x +   3*x^2 +    x^3;
   15*x +  15*x^2 +  6*x^3 +    x^4;
  105*x + 105*x^2 + 45*x^3 + 10*x^4 + x^5;
  ...
Triangle of coefficients begins as:
  0;
  1,  0;
  1,  1    0;
  1,  3,   3     0;
  1,  6,  15,   15      0;
  1, 10,  45,  105,   105      0;
  1, 15, 105,  420,   945,   945       0;
  1, 21, 210, 1260,  4725, 10395,  10395       0;
  1, 28, 378, 3150, 17325, 62370, 135135, 135135    0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Bessel Polynomial

Essentially the same as A001498 (the main entry).

Cf. A000085, A000806, A001464, A001515.

A104548:= func< n,k | k eq n select 0 else Binomial(n-1,k)*Factorial(n+k-1)/(2^k*Factorial(n-1)) >;
[A104548(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 02 2023

T[n_, k_]:= If[k==n, 0, Binomial[n-1,k]*(n+k-1)!/(2^k*(n-1)!)];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 02 2023 *)

def A104548(n,k): return 0 if (k==n) else binomial(n-1,k)*factorial(n+k-1)/(2^k*factorial(n-1))
flatten([[A104548(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 02 2023

From G. C. Greubel, Jan 02 2023: (Start)
T(n, k) = binomial(n-1,k)*(n+k-1)!/(2^k*(n-1)!), with T(n, n) = 0.
Sum_{k=0..n} T(n, k) = A001515(n-1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000806(n-1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000085(n-1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A001464(n-1). (End)

T(0, 0) = 0 prepended by G. C. Greubel, Jan 02 2023

A107103 Column 1 of triangle A107102, which is the matrix inverse of A100862.

Original entry on oeis.org

1, -3, 12, -67, 495, -4596, 51583, -680037, 10306152, -176591665, 3376207461, -71258869848, 1645797382177, -41289185878227, 1118136109703460, -32509366699961371, 1010047705084550823, -33397162601828122332, 1170937325957822375167, -43391988679237460897205

Offset: 0

Paul D. Hanna, May 21 2005

sign

Column 0 of triangle A107102 is signed A001515 (Bessel polynomial).

Cf. A100862, A001515, A043301, A107102, A107104.

{a(n)=local(X=x+x^2*O(x^n),Y=y+y^2*O(y^n));(matrix(n+2,n+2,m,k,if(m>=k, (m-1)!*polcoeff(polcoeff(exp(X+Y*X^2/2+X*Y),m-1,x),k-1,y)))^-1)[n+2,2]}

|a(n)| = 2*A001515(n)-A001515(n-1). - Vladeta Jovovic, Aug 10 2006

A144576 E.g.f.: exp(1-sqrt(1-2x-4x^2)).

Original entry on oeis.org

1, 1, 6, 31, 301, 3426, 51751, 926731, 19691106, 479961901, 13256384851, 408621822126, 13915350562081, 518741273626681, 21013220503491126, 919071064063596151, 43167975952565245501, 2167078807061679282306, 115795155400715170458631, 6561750899663711363984851

Offset: 0

N. J. A. Sloane, Jan 07 2009

nonn

Robert Israel, Table of n, a(n) for n = 0..377

Cf. A001515, A144575.

f:= gfun:-rectoproc({a(n+5) = 64*(n+3)*(n+2)*(n+1)*a(n)+48*(n+3)*(n+2)*a(n+1)+4*(n+3)*(4*n^2+12*n+11)*a(n+2)+(12*n^2+60*n+73)*a(n+3)-(2*n+1)*a(n+4), a(0) = 1, a(1) = 1, a(2) = 6, a(3) = 31, a(4) = 301}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Dec 31 2019

With[{nn=20},CoefficientList[Series[Exp[1-Sqrt[1-2x-4x^2]],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Apr 30 2012 *)

a(n) ~ sqrt(5-sqrt(5))*(1+sqrt(5))^n*n^n/(2*n*exp(n-1)). - Vaclav Kotesovec, Jun 26 2013

D-finite with recurrence: a(n) +(-2*n+3)*a(n-1) +(-4*n^2+16*n-13)*a(n-2) +4*(-2*n+3)*a(n-3) -16*(n-1)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 23 2020

A302285 Expansion of 1/(1 - x - x/(1 - 2x - x/(1 - 3x - x/(1 - 4x - x/(1 - 5x - x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 2, 7, 33, 185, 1170, 8121, 60846, 486753, 4125852, 36846557, 345205559, 3381126995, 34524194712, 366635359887, 4041180951473, 46149726728969, 545161967955668, 6652026230285141, 83730953689450825, 1085924693069106823, 14494802798426546660, 198918641942013097723

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

a(n) is the number of paths from (0,0) to (2n,0) on or above the x-axis with steps U=(1,1), D=(1,-1), and L=(2,0), where the level steps L at height k have k+1 colors for all k>=0. - Alexander Burstein, Apr 10 2025

			G.f. A(x) = 1 + 2*x + 7*x^2 + 33*x^3 + 185*x^4 + 1170*x^5 + 8121*x^6 + 60846*x^7 + 486753*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..549
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.

Cf. A001515, A006318, A006351, A006789, A258173, A295289.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, b(x-1, y-1)+b(x-1, y+1)+b(x-2, y)*(y+1)))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2025

nmax = 22; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x, 1 - (k + 1) x, {k, 1, nmax}]), {x, 0, nmax}], x]

cp's OEIS Frontend

A105748 Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a collection of n (possibly empty) sets, each with at most 2 elements.

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Maple

Mathematica

PARI

Formula

A107102 Matrix inverse of A100862.

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PARI

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A143968 Denominators of numbers with g.f. exp(1-(1-x)^(1/2)).

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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.

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Maple

Mathematica

Formula

A144513 a(n) = Sum_{k=0..n} (n+k+2)!/((n-k)!*k!*2^k).

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Maple

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A144514 a(n) = Sum_{k=0..n} (n+k+3)!/((n-k)!*k!*2^k).

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Maple

Mathematica

PARI

Formula

A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

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Magma

Mathematica

SageMath

Formula

Extensions

A107103 Column 1 of triangle A107102, which is the matrix inverse of A100862.

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A144513 a(n) = Sum_{k=0..n} (n+k+2)!/((n-k)!k!2^k).

A144514 a(n) = Sum_{k=0..n} (n+k+3)!/((n-k)!k!2^k).

A144576 E.g.f.: exp(1-sqrt(1-2x-4x^2)).

A302285 Expansion of 1/(1 - x - x/(1 - 2x - x/(1 - 3x - x/(1 - 4x - x/(1 - 5x - x/(1 - ...)))))), a continued fraction.