A001515 -id:A001515 - cp's OEIS frontend

A105749 Number of ways to use the elements of {1,...,k}, 0 <= k <= 2n, once each to form a sequence of n sets, each having 1 or 2 elements.

1, 2, 14, 222, 6384, 291720, 19445040, 1781750880, 214899027840, 33007837322880, 6290830003852800, 1456812592995513600, 402910665227270323200, 131173228963370155161600, 49656810289225281849907200, 21628258853895305337293568000, 10739534026001485514941587456000

Offset: 0

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

Equivalently, number of sequences of n labeled items such that each item occurs just once or twice. - David Applegate, Dec 08 2008

Also, number of assembly trees for a certain star graph, see Vince-Bona, Theorem 4. - N. J. A. Sloane, Oct 08 2012

			a(2) = 14 = |{ ({1},{2}), ({2},{1}), ({1},{2,3}), ({2,3},{1}), ({2},{1,3}), ({1,3},{2}), ({3},{1,2}), ({1,2},{3}), ({1,2},{3,4}), ({3,4},{1,2}), ({1,3},{2,4}), ({2,4},{1,3}), ({1,4},{2,3}), ({2,3},{1,4}) }|.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Samuele Giraudo and Yannic Vargas, Operads of packed words, quotients, and monomial bases, Proc. 37th Conf. Formal Power Series Alg. Comb., Sém. Lotharingien Combin. (2025) Vol. 93B Art. No. 82. See p. 9.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
Index entries for related partition-counting sequences

Cf. A001515, A003011, A143990.
Replace "sets" by "lists": A099022.
Column n=2 of A181731.

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

a:= n-> add(binomial(n, k)*(n+k)!/2^k, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2012

f[n_]:= Sum[Binomial[n,k]*(n+k)!/2^k, {k,0,n}]; Table[f[n], {n,0,20}]

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

a(n) = Sum_{k=0..n} C(n,k) * (n+k)! / 2^k.
a(n) = n! * A001515(n).
A003011(n) = Sum_{k=0..n} C(n, k)*a(k).
a(n) = Gamma(n+1)*Hypergeometric2F0([-n, n+1], [], -1/2). - Peter Luschny, Jul 29 2014
a(n) ~ sqrt(Pi) * 2^(n + 1) * n^(2*n + 1/2) / exp(2*n - 1). - Vaclav Kotesovec, Nov 27 2017
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = n*(2*n-1)*a(n-1) + n*(n-1)*a(n-2).
a(n) = e * sqrt(2/Pi) * n! * BesselK(n+1/2, 1).
a(n) = ((2*n)!/2^n) * Hypergeometric1F1(-n, -2*n, 2).
G.f.: (-2/x) * Integrate_{t=0..oo} exp(-t)/((t+1)^2 - 1 - 2/x) dt.
G.f.: e*( exp(-sqrt(1 + 2/x)) * ExpIntegralEi(-1 + sqrt(1 + 2/x)) - exp(sqrt(1 + 2/x)) * ExpIntegralEi(-1 - sqrt(1 + 2/x)) )/sqrt(x^2 + 2*x).
E.g.f.: ((1-x)/x) * Hypergeometric1F1(1, 3/2, -(1-x)^2/(2*x)).
E.g.f.: (1/(1-x))*Hypergeometric2F0([1, 1/2]; []; 2*x/(1-x)^2). (End)

More terms from Robert G. Wilson v, Apr 23 2005

A144505 Triangle read by rows: coefficients of polynomials arising from the recurrence A[n](x) = A[n-1]'(x)/(1-x) with A[0] = exp(x).

Original entry on oeis.org

1, 1, -1, 2, 1, -5, 7, -1, 9, -30, 37, 1, -14, 81, -229, 266, -1, 20, -175, 835, -2165, 2431, 1, -27, 330, -2330, 9990, -24576, 27007, -1, 35, -567, 5495, -34300, 137466, -326515, 353522, 1, -44, 910, -11522, 97405, -561386, 2148139, -4976315, 5329837

Offset: 0

N. J. A. Sloane, Dec 14 2008

			The first few polynomials P[n] (n >= 0) are:
  P[0] = 1;
  P[1] = 1;
  P[2] = -x +2;
  P[3] =  x^2 -5*x +7;
  P[4] = -x^3 + 9*x^2 - 30*x +37;
  P[5] =  x^4 -14*x^3 + 81*x^2 - 229*x +266;
  P[6] = -x^5 +20*x^4 -175*x^3 + 835*x^2 -2165*x +2431;
  P[7] =  x^6 -27*x^5 +330*x^4 -2330*x^3 +9990*x^2 -24576*x +27007;
...
Triangle of coefficients begins:
   1;
   1;
  -1,   2;
   1,  -5,    7;
  -1,   9,  -30,     37;
   1, -14,   81,   -229,    266;
  -1,  20, -175,    835,  -2165,    2431;
   1, -27,  330,  -2330,   9990,  -24576,   27007;
  -1,  35, -567,   5495, -34300,  137466, -326515,   353522;
   1, -44,  910, -11522,  97405, -561386, 2148139, -4976315, 5329837;
...

Seiichi Manyama, Rows n = 0..140, flattened
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See pp. 42, 63.
N. J. A. Sloane, Rows 0 through 25 of the triangle, together with the corresponding polynomials P[n](x).

Columns give A001515 (really A144301), A144498, A001514, A144506, A144507.
Row sums give A001147.
Alternating row sums give A043301.

R:=PowerSeriesRing(Integers(), 50);
f:= func< n,x | x^n*(&+[Binomial(n,j)*Factorial(n+j)*(1-1/x)^(n-j)/(2^j*Factorial(n)) : j in [0..n]]) >;
T:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[1] cat [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 02 2023

A[0]:=exp(x);
P[0]:=1;
for n from 1 to 12 do
A[n]:=sort(simplify( diff(A[n-1],x)/(1-x)));
P[n]:=sort(simplify(A[n]*(1-x)^(2*n-1)/exp(x)));
t1:=simplify(x^(degree(P[n],x))*subs(x=1/x,P[n]));
t2:=series(t1,x,2*n+3);
lprint(P[n]);
lprint(seriestolist(t2));
od:

f[n_, x_]:= x^n*Sum[((n+j)!/((n-j)!*j!*2^j))*(1-1/x)^(n-j), {j,0,n}];
t[n_, k_]:= Coefficient[Series[f[n,x], {x,0,30}], x, k];
Join[{1}, Table[t[n,k], {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Oct 02 2023 *)

P. = PowerSeriesRing(QQ, 50)
def f(n,x): return x^n*sum(binomial(n,j)*rising_factorial(n+1,j)*(1-1/x)^(n-j)/2^j for j in range(n+1))
def T(n,k): return P( f(n,x) ).list()[k]
[1] + flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 02 2023

Let A[0](x) = exp(x), A[n](x) = A[n-1]'(x)/(1-x) for n>0 and let P[n](x) = A[n](x)*(1-x)^(2n-1)/exp(x). Row n of triangle gives coefficients of P[n](x) with exponents of x in decreasing order.
From Vladeta Jovovic, Dec 15 2008: (Start)
P[n] = Sum_{k=0..n} ((n+k)!/((n-k)!*k!*2^k)) * (1-x)^(n-k).
E.g.f.: exp((1-x)*(1-sqrt(1-2*y)))/sqrt(1-2*y). (End)

A015735 Row sums of triangle A004747.

Original entry on oeis.org

1, 3, 17, 145, 1661, 23931, 415773, 8460257, 197360985, 5192853011, 152137882601, 4911873672113, 173268075672277, 6630323916472075, 273555262963272501, 12105084133976359361, 571897644855277242673, 28731255563712689630627, 1529450942687399074134465

Offset: 1

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A001515, A004747, A016036, A028575.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-3*x)^(1/3)) - 1 ))); // G. C. Greubel, Oct 02 2023

a[1]=1; a[n_]:= 1 +(n-1)!*Sum[Binomial[k, n-m-k]*Binomial[k+n-1,n-1]*(-1/3)^(n-m-k)/(m-1)!, {m,n}, {k,n-m}]; Table[a[n], {n,20}] (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
Rest@With[{m=30}, CoefficientList[Series[Exp[1-Surd[1-3*x,3]] -1, {x, 0,m}], x]*Range[0,m]!] (* G. C. Greubel, Oct 02 2023 *)

a(n):=if n=1 then 1 else (n-1)!*sum(sum(binomial(k,n-m-k)* (-1/3)^(n-m-k)*binomial(k+n-1,n-1),k,1,n-m)/(m-1)!,m,1,n)+1; /* Vladimir Kruchinin, Aug 08 2010 */

def A015735_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-3*x)^(1/3)) -1 ).egf_to_ogf().list()
a=A015735_list(40); a[1:] # G. C. Greubel, Oct 02 2023

E.g.f.: exp(1-(1-3*x)^(1/3)) - 1, if one takes a(0)=0.
a(n) = 6*(n-2)*a(n-1) - (3*n-8)*(3*n-7)*a(n-2) + a(n-3), a(0)=1, a(1)=1, a(2)=3.
a(n) = 1 + (n-1)!*Sum_{m=1..n} ( Sum_{k=1..n-m} C(k, n-m-k)*C(k+n-1, n-1)*(-1/3)^(n-m-k) ) / (m-1)!, n > 1. - Vladimir Kruchinin, Aug 08 2010
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^2*d/dx. Cf. A001515, A016036 and A028575. - Peter Bala, Nov 25 2011
E.g.f. with offset 0: exp(1-(1-3*x)^(1/3))/(1-3*x)^(2/3). - Sergei N. Gladkovskii, Jul 07 2012.
a(n) ~ sqrt(2*Pi)*3^(n-1)*exp(1-n)*n^(n-5/6)/Gamma(2/3) * (1-sqrt(3)*Gamma(2/3)^2/(2*Pi*n^(1/3))). - Vaclav Kotesovec, Aug 10 2013
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-3)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/3,n)/k!. (End)

A016036 Row sums of triangle A000369.

Original entry on oeis.org

1, 4, 31, 361, 5626, 109951, 2585269, 71066626, 2236441141, 79289379361, 3127129674736, 135802922499949, 6439320471558781, 331026965612789356, 18338413238239145731, 1089132347371148170381, 69033182553940825258594, 4651256393180943757676371

Offset: 1

Wolfdieter Lang

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Sequences with e.g.f. exp(1-(1-m*x)^(1/m)) - 1: A000012 (m=1), A001515 (m=2), A015735 (m=3), this sequence (m=4), A028575 (m=5), A028844 (m=6).

Cf. A000369.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-4*x)^(1/4)) -1 ))); // G. C. Greubel, Oct 02 2023

a[n_, m_] /; (n>= m>= 1):= a[n, m]= (4*(n-1)-m)*a[n-1,m] + a[n-1,m-1]; a[n_, m_] /; n,0]= 0; a[1,1] = 1; a[n]:= Sum[a[n,m], {m, n}]; Table[a[n], {n,20}] (* Jean-François Alcover, Feb 28 2013 *)
With[{nn=20},CoefficientList[Series[Exp[1-Surd[1-4x,4]]-1,{x,0,nn}],x] Range[0,nn]!]//Rest (* Harvey P. Dale, Apr 20 2016 *)

a(n):=((n-1)!*sum((sum(binomial(n+k-1,n-1)*sum(binomial(j,n-m-3*k+2*j)*binomial(k,j)*3^(-n+m+3*k-j)*2^(n-m-5*k+3*j)*(-1)^(n-m-k),j,0,k),k,1,n-m))/(m-1)!,m,1,n-1))+1; /* Vladimir Kruchinin, Oct 18 2011 */

def A016036_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-4*x)^(1/4)) -1 ).egf_to_ogf().list()
a=A016036_list(40); a[1:] # G. C. Greubel, Oct 02 2023

E.g.f.: exp(1 - (1-4*x)^(1/4)) - 1.
a(n) = 6*(2*n-5)*a(n-1) - 3*(16*n^2-96*n+145)*a(n-2) + 2*(4*n-15)*(2*n-7)*(4*n-13)*a(n-3) + a(n-4), n >= 4; a(0) = 1, a(1) = 1, a(2) = 4, a(3) = 31.
a(n) = 1 + (n-1)!*Sum_{m=1..n-1} ( Sum_{k=1..n-m} binomial(n+k-1,n-1) * ( Sum_{j=0..k} binomial(j,n-m-3*k+2*j)*binomial(k,j)*3^(-n+m+3*k-j)*2^(n-m-5*k+3*j)*(-1)^(n-m-k) ) )/(m-1)!. - Vladimir Kruchinin, Oct 18 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^3*d/dx. Cf. A001515, A015735 and A028575. - Peter Bala, Nov 25 2011
a(n) ~ 2^(2*n-3/2)*n^(n-3/4)*exp(1-n)*sqrt(Pi)/Gamma(3/4) * (1 - Gamma(3/4)/(n^(1/4)*sqrt(Pi)) + Gamma(3/4)^2/(4*sqrt(n/2)*Pi)). - Vaclav Kotesovec, Aug 10 2013
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 4^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-4)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/4,n)/k!. (End)

A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 3, 3, 0, 0, 0, 1, 6, 15, 15, 0, 0, 0, 0, 1, 10, 45, 105, 105, 0, 0, 0, 0, 0, 1, 15, 105, 420, 945, 945, 0, 0, 0, 0, 0, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 0, 0, 0, 0, 0, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 0, 0, 0, 0, 0

Offset: 0

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

Although this entry is the last of the versions of the underlying triangle to be added to the OEIS, for some applications it is the most important.
Row n has 2n+1 entries.
A001498 has a b-file.

			Triangle begins:
  1
  0 1 1
  0 0 1 3 3
  0 0 0 1 6 15 15
  0 0 0 0 1 10 45 105 105
  0 0 0 0 0  1 15 105 420  945  945
  0 0 0 0 0  0  1  21 210 1260 4725 10395 10395
  ...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144299, A278990.
Row sums give A001515, column sums A000085.
Other versions of this triangle are given in A001497, A001498, A111924 and A100861.
See A144385 for a generalization.

a144331 n k = a144331_tabf !! n !! k
a144331_row n = a144331_tabf !! n
a144331_tabf = iterate (\xs ->
  zipWith (+) ([0] ++ xs ++ [0]) $ zipWith (*) (0:[0..]) ([0,0] ++ xs)) [1]
-- Reinhard Zumkeller, Nov 24 2014

A144331:= func< n,k | k le n-1 select 0 else Factorial(k)/(2^(k-n)*Factorial(k-n)*Factorial(2*n-k)) >;
[A144331(n,k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Oct 04 2023

Flatten[Table[PadLeft[Table[(n+k)!/(2^k*k!*(n-k)!), {k,0,n}], 2*n+1, 0], {n,0,12}]] (* Jean-François Alcover, Oct 14 2011 *)

def A144331(n, k): return 0 if kA144331(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Oct 04 2023

E.g.f.: Sum_{n >= 0} Sum_{k = 0..2n} b(n,k) y^n * x^k/k! = exp(x*y*(1 + x/2)).
b(n, k) = 2^(n-k)*k!/((2*n-k)!*(k-n)!).
Sum_{k=0..2*n} b(n, k) = A001515(n).
Sum_{n >= 0} b(n, k) = A000085(k).
From G. C. Greubel, Oct 04 2023: (Start)
T(n, k) = 0 for 0 <= k <= n-1, otherwise T(n, k) = k!/(2^(k-n)*(k-n)!*(2*n-k)!) for n <= k <= 2*n.
Sum_{k=0..2*n} (-1)^k * T(n, k) = A278990(n). (End)

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

A065919 Bessel polynomial y_n(4).

Original entry on oeis.org

1, 5, 61, 1225, 34361, 1238221, 54516085, 2836074641, 170218994545, 11577727703701, 880077524475821, 73938089783672665, 6803184337622361001, 680392371852019772765, 73489179344355757819621, 8525425196317119926848801, 1057226213522667226687070945

Offset: 0

N. J. A. Sloane, Dec 08 2001

Main diagonal of A143411. - Peter Bala, Aug 14 2008

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

Harry J. Smith, Table of n, a(n) for n = 0..100
W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A001517, A001518.
Cf. A143411 (main diagonal), A143412.
Polynomial coefficients are in A001498.

A065919:= func< n | (&+[Binomial(n,k)*Factorial(n+k)*2^k/Factorial(n): k in [0..n]]) >;
[A065919(n): n in [0..30]]; // G. C. Greubel, Oct 05 2023

seq(simplify(2^n*KummerU(-n,-2*n,1/2)), n=0..16); # Peter Luschny, May 10 2022

Table[Sum[(n+k)!*2^k/((n-k)!*k!), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)

for (n=0, 100, if (n>1, a=4*(2*n - 1)*a1 + a2; a2=a1; a1=a, if (n, a=a1=5, a=a2=1)); write("b065919.txt", n, " ", a) ) \\ Harry J. Smith, Nov 04 2009

a(n) = sum(k=0,n, (n+k)!*2^k/((n-k)!*k!) ); \\ Joerg Arndt, May 17 2013

def A065919(n): return sum(binomial(n,k)*factorial(n+k)*2^k/factorial(n) for k in range(n+1))
[A065919(n) for n in range(31)] # G. C. Greubel, Oct 05 2023

y_n(x) = Sum_{k=0..n} (n+k)!*(x/2)^k/((n-k)!*k!).
From Peter Bala, Aug 14 2008: (Start)
Recurrence relation: a(0) = 1, a(1) = 5, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A143412(n) satisfies the same recurrence relation.
1/sqrt(e) = 1 - 2*Sum_{n = 0..inf} (-1)^n/(a(n)*a(n+1)) = 1 - 2*( 1/(1*5) - 1/(5*61) + 1/(61*1225) - ... ). (End)
G.f.: 1/Q(0), where Q(k)= 1 - x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = exp(1/4)/sqrt(2*Pi)*BesselK(n+1/2,1/4). - Gerry Martens, Jul 22 2015
a(n) ~ 2^(3*n+1/2) * n^n / exp(n-1/4). - Vaclav Kotesovec, Jul 22 2015
From Peter Bala, Apr 12 2017: (Start)
a(n) = 1/n!*Integral_{x = 0..inf} x^n*(1 + 2*x)^n dx.
E.g.f.: d/dx( exp(x*c(2*x)) ) = 1 + 5*x + 61*x^2/2! + 1225*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
G.f.: (1/(1-x))*hypergeometric2f0(1,1/2; - ; 8*x/(1-x)^2). - G. C. Greubel, Aug 16 2017
a(n) = 2^n*KummerU(-n, -2*n, 1/2). - Peter Luschny, May 10 2022

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 *)

A028844 Row sums of triangle A013988.

Original entry on oeis.org

1, 6, 71, 1261, 29906, 887751, 31657851, 1318279586, 62783681421, 3365947782611, 200610405843926, 13157941480889921, 941848076798467801, 73060842413607398806, 6105266987293752470991, 546770299628690541571901, 52244284936267317229542466, 5305131708827069245129523591

Offset: 1

Wolfdieter Lang

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Sequences with e.g.f. exp(1-(1-m*x)^(1/m)) - 1: A000012 (m=1), A001515 (m=2), A015735 (m=3), A016036 (m=4), A028575 (m=5), this sequence (m=6).

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-6*x)^(1/6)) -1 ))); // G. C. Greubel, Oct 03 2023

With[{nn=20},Rest[CoefficientList[Series[Exp[1-(1-6x)^(1/6)]-1,{x,0,nn}], x]Range[0,nn]!]] (* Harvey P. Dale, Feb 02 2012 *)

def A028844_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(1-(1-6*x)^(1/6)) -1 ).egf_to_ogf().list()
a=A028844_list(40); a[1:] # G. C. Greubel, Oct 03 2023

E.g.f.: exp(1 - (1-6*x)^(1/6)) - 1.
D-finite with recurrence: a(n) = 15*(2*n-7)*a(n-1) +5*(72*n^2-576*n+1169)*a(n-2) +45*(2*n-9)*(24*n^2-216*n+497)*a(n-3) -20*(324*n^4-6480*n^3+48735*n^2-163350*n+205877)*a(n-4) +12*(6*n-35)*(6*n-31)*(3*n-16)*(2*n-11)*(3*n-17)*a(n-5) +a(n-6). - R. J. Mathar, Jan 28 2020
From Seiichi Manyama, Jan 20 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-6)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/6,n)/k!. (End)

A143991 Numerators of numbers with g.f. exp(1-(1-x)^(1/2)).

Original entry on oeis.org

1, 1, 1, 7, 37, 133, 2431, 27007, 176761, 5329837, 12994393, 866792053, 5213746711, 841146804577, 10532583170701, 569600638022431, 16539483668991901, 3333075288160853, 16955228098102446847, 932411737877492011, 10996483739066355827053, 66024590609554132070857

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. A143968 (denominators), A001515.

Cf. A381845.

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

CoefficientList[Series[Exp[1-Sqrt[1-x]],{x,0,30}],x]//Numerator (* Harvey P. Dale, Nov 23 2024 *)

Conjecture: a(n) = numerator( (e/Pi)*Integral_{x=-oo..+oo} cos(x)/(1 + x^2)^n dx ) for n > 0. See A381845 for the numerators of this integral. - Stefano Spezia, Mar 12 2025

cp's OEIS Frontend

A105749 Number of ways to use the elements of {1,...,k}, 0 <= k <= 2n, once each to form a sequence of n sets, each having 1 or 2 elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions

A144505 Triangle read by rows: coefficients of polynomials arising from the recurrence A[n](x) = A[n-1]'(x)/(1-x) with A[0] = exp(x).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A015735 Row sums of triangle A004747.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

SageMath

Formula

A016036 Row sums of triangle A000369.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

SageMath

Formula

A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

SageMath

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A065919 Bessel polynomial y_n(4).

Views