A002212 -id:A002212 - cp's OEIS frontend

A371486 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^4.

1, 4, 30, 260, 2465, 24796, 260008, 2811216, 31117240, 350890260, 4016744586, 46556054072, 545273713228, 6443442857024, 76727957438650, 919796418086076, 11091249210406816, 134439965189940176, 1637160457090585016, 20019920157735604796, 245733987135102838131

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A002212, A270386, A371483.

Cf. A118971, A349332.

a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(5*k+3, k)/(k+1));

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+3,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^4 ).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349332.

A384747 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,..,k}, and no nodes have the same weight as their parent node.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 6, 0, 1, 11, 15, 16, 0, 1, 26, 39, 43, 44, 0, 1, 63, 110, 123, 127, 128, 0, 1, 153, 308, 358, 371, 375, 376, 0, 1, 376, 869, 1046, 1096, 1109, 1113, 1114, 0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346, 0, 1, 2317, 7238, 9283, 9904, 10084, 10134, 10147, 10151, 10152

Offset: 0

John Tyler Rascoe, Jun 09 2025

			Triangle begins:
    k=0  1    2     3     4     5     6     7     8     9
 n=0 [1]
 n=1 [0, 1]
 n=2 [0, 1,   2]
 n=3 [0, 1,   5,    6]
 n=4 [0, 1,  11,   15,   16]
 n=5 [0, 1,  26,   39,   43,   44]
 n=6 [0, 1,  63,  110,  123,  127,  128]
 n=7 [0, 1, 153,  308,  358,  371,  375,  376]
 n=8 [0, 1, 376,  869, 1046, 1096, 1109, 1113, 1114]
 n=9 [0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346]
...
T(3,3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)

Cf. A051286 (column k=2), A382096 (column k=3), A384748 (main diagonal).

Cf. A000108, A002212, A143330, A384613, A384685.

b(i,j,k,N) = {if(k>N,1, 1/( 1  - sum(u=1,j, if(u==i,0,x^u * b(u,j,k+1,N-u+1)))))}
Gx(k,N) = {my(x='x+O('x^(N+1))); Vec(1/(1 - sum(i=1,k, b(i,k,1,N)*x^i)))}
T(max_row) = { my( N = max_row+1, v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Gx(k,N)~])); vector(N, n, vector(n, k, v[n, k]))}
T(9)

T(n,k) = T(n,n) for k > n.

A045445 Number of nonisomorphic systems of catafusenes for the unsymmetrical schemes (group C_s) with two appendages (see references for precise definition).

Original entry on oeis.org

0, 1, 6, 29, 132, 590, 2628, 11732, 52608, 237129, 1074510, 4893801, 22395420, 102943815, 475139070, 2201301575, 10234016880, 47731093715, 223273611810, 1047265325255, 4924606035900, 23211459517120, 109642275853176, 518959629394294, 2460993383491632, 11691102386417575

Offset: 1

N. J. A. Sloane

Number of 3-Motzkin paths of length n (i.e., lattice paths from (0,0) to (n,0) that do not go below the line y = 0 and consist of steps U = (1,1), D = (1,-1) and three types of steps H = (1,0)) that start with a U step. Example: a(4) = 29 because we have UDUD, UUDD, 9 UDHH paths, 9 UHDH paths and 9 UHHD paths. - Emeric Deutsch, Mar 26 2004

Here, n is the total number of hexagons in the system, which is usually denoted by h in most of the references below. In Cyvin, Brunvoll, and Cyvin (1992), Table 1, p. 28, it seems that the rooted hexagon is "distinguished", and the sequence is shifted by 1. - Petros Hadjicostas, May 26 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Harary-Read numbers for catafusenes: Complete classification according to symmetry, Journal of Mathematical Chemistry 9(1) (1992), 19-31; see Table 1 (p. 28).
S. J. Cyvin and J. Brunvoll, Generating functions for the Harary-Read numbers classified according to symmetry, Journal of Mathematical Chemistry 9(1) (1992), 33-38.
B. N. Cyvin, E. Brendsdal, J. Brunvoll, and S. J. Cyvin, A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337; see Eqs (10) and (13) on p. 1330.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180; see "Two Appendages" in Fig. 1 (p. 1176) for the unsymmetrical case (group C_s).
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. 17(2) (1970), 1-13.
Eric Weisstein's World of Mathematics, Fusene.
Wikipedia, Molecular symmetry.
Wikipedia, Point groups in three dimensions.
Wikipedia, Polyhex (mathematics).
Wikipedia, Schoenflies notation.

Cf. A002212, A045829 (auto-convolution), A002057.

a := n -> binomial(2*n+2,n+1)/(n+2) + add(binomial(2*k,k)*binomial(n-1,k-1)*(3*k-2*n-3)/(n-k+1)/(k+1),k=1..n): 0,seq(a(n),n=2..23);
# Alternative:
a := n -> (2*(n - 1)/(n + 2))*(binomial(2*n, n) / (n + 1))*hypergeom([-n-2, -n+2], [-n + 1/2], -1/4): seq(simplify(a(n)), n = 1..26); # Peter Luschny, Oct 23 2022

a[n_] = Binomial[2n+2, n+1]/(n+2) + Sum[Binomial[2k, k]*Binomial[n-1, k-1]*(3k-2n-3)/(n-k+1)/(k+1), {k, 1, n}];
a /@ Range[23] (* Jean-François Alcover, Jul 13 2011, after Maple *)
Table[SeriesCoefficient[(1/2)*(7*x^2-6*x+1+(3*x-1)*Sqrt[5*x^2-6*x+1])/x^2,{x,0,n}],{n,1,23}] (* Vaclav Kotesovec, Oct 08 2012 *)

x='x+O('x^66); concat([0],Vec((1/2)*(7*x^2-6*x+1+(3*x-1)*sqrt(5*x^2-6*x+1))/x^2)) \\ Joerg Arndt, May 04 2013

G.f.: (1/2)*(7*x^2 - 6*x + 1 + (3*x-1)*sqrt(5*x^2-6*x+1))/x^2. - Vladeta Jovovic, Jul 19 2001
a(n) = A002212(n+1) - 3*A002212(n). Convolution of A002212 without the first term with itself. - Emeric Deutsch, Jul 24 2002
a(n) = binomial(2n+2, n+1)/(n+2) + Sum_{k=1..n} binomial(2k, k)*binomial(n-1, k-1)*(3k-2n-3)/((n-k+1)*(k+1)) (n >= 2). - Emeric Deutsch, Mar 26 2004
Recurrence: (n-2)*(n+2)*a(n) = 3*(n-1)*(2*n-1)*a(n-1) - 5*(n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
a(n) = (2/(n+1))*Sum_{m=0..n-1} C(n+1,m)*C(2*n-2*m+2,n-m-1). - Vladimir Kruchinin Oct 18 2022
Let h(n) = hypergeom([-n-2, -n+2], [-n+1/2], -1/4) then a(n) = A002057(n-2)*h(n) = (2*(n-1)/(n+2))*CatalanNumber(n)*h(n). - Peter Luschny, Oct 23 2022

More terms from Vladeta Jovovic, Jul 19 2001

A371483 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 18, 124, 933, 7446, 61943, 531348, 4666425, 41751325, 379230711, 3487769871, 32414437521, 303950138604, 2872137458010, 27322233357964, 261446381792670, 2514851398148595, 24303030755342128, 235841264063844258, 2297278004837062317

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A002212, A270386, A371486.

Cf. A006632, A349331.

a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(4*k+2, k)/(k+1));

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+2,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^3 ).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A349331.

A385563 Expansion of 1/((1-x) * (1-5*x))^(3/2).

Original entry on oeis.org

1, 9, 60, 360, 2055, 11403, 62132, 334260, 1781415, 9425295, 49581576, 259601004, 1353939405, 7038232425, 36484340400, 188665670880, 973545780195, 5014258620075, 25783103206100, 132378800689800, 678768332410245, 3476164133573505, 17782899991147500

Offset: 0

Seiichi Manyama, Aug 19 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000

Partial sums of A383254.
Cf. A331516, A385716.
Cf. A385728, A385813.
Cf. A002212, A383949.

Module[{a, n}, RecurrenceTable[{a[n] == ((6*n+3)*a[n-1] - 5*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 9}, a, {n, 0, 25}]] (* Paolo Xausa, Aug 21 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(3/2))

n*a(n) = (6*n+3)*a(n-1) - 5*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A002212(n+1).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (3/2)^k * (-5/6)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(n) * 5^(n + 3/2) / (4*sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

A002215 Number of polyhexes with n hexagons, having reflectional symmetry (see Harary and Read for precise definition).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 36, 72, 137, 274, 543, 1086, 2219, 4438, 9285, 18570, 39587, 79174, 171369, 342738, 751236, 1502472, 3328218, 6656436, 14878455, 29756910, 67030785, 134061570, 304036170, 608072340, 1387247580, 2774495160

Offset: 1

N. J. A. Sloane

nonn

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).

T. D. Noe, Table of n, a(n) for n = 1..200.
J. Brunvoll, S. J. Cyvin, and B. N. Cyvin, Isomer enumeration of polygonal systems..., J. Molec. Struct. (Theochem), 364 (1996), 1-13, Table 10.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Cf. A002212.

From Emeric Deutsch, Mar 14 2004: (Start)
G.f.: z + (1+2*z)*U(z^2) where U(z) = (1 - 3*z - sqrt(1-6*z+5*z^2))/(2*z) (eq. (16) in the Harary-Read paper).
a(2n) = A002212(n), n >= 1; a(2n+1) = 2*A002212(n), n >= 1. (End)

More terms from Emeric Deutsch, Mar 14 2004

A108198 Triangle read by rows: T(n,k) = binomial(2k+2,k+1)*binomial(n,k)/(k+2) (0 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 1, 6, 15, 14, 1, 8, 30, 56, 42, 1, 10, 50, 140, 210, 132, 1, 12, 75, 280, 630, 792, 429, 1, 14, 105, 490, 1470, 2772, 3003, 1430, 1, 16, 140, 784, 2940, 7392, 12012, 11440, 4862, 1, 18, 180, 1176, 5292, 16632, 36036, 51480, 43758, 16796, 1, 20, 225

Offset: 0

Emeric Deutsch, Jun 15 2005, Mar 30 2007

Also, with offset 1, triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and ending at the point (2k,0) (1 <= k <= n). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. For example, T(3,2)=4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDDL.

This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) of degree n and shift 1 for the Catalan sequence A000108. See A098474 for a similar comment. - Wolfdieter Lang, Jun 25 2019

			Triangle begins:
1;
1,  2;
1,  4,  5;
1,  6, 15,  14;
1,  8, 30,  56,  42;
1, 10, 50, 140, 210, 132;
1, 12, 75, 280, 630, 792, 429;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Mirror image of A126181.

Cf. A000108, A002212, A026388, A098474.

T:=(n,k)->binomial(2*k+2,k+1)*binomial(n,k)/(k+2): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
h := n -> simplify(hypergeom([3/2, -n], [3], -x)):
seq(print(seq(4^k*coeff(h(n), x, k), k=0..n)), n=0..9); # Peter Luschny, Feb 03 2015

Flatten[Table[Binomial[2k+2,k+1] Binomial[n,k]/(k+2),{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 20 2013 *)

def A108198(n,k):
    return (-1)^k*catalan_number(k+1)*rising_factorial(-n,k)/factorial(k)
for n in range(7): [A108198(n,k) for k in (0..n)] # Peter Luschny, Feb 05 2015

Sum of row n = A002212(n+1).
T(n,n) = Catalan(n+1) (A000108).
Sum_{k=1..n} k*T(n,k) = A026388(n).
With offset 1, T(n,k) = c(k)*binomial(n-1,k-1), where c(j) = binomial(2j,j)/(j+1) is a Catalan number (A000108).
G.f.: G-1, where G=G(t,z) satisfies G = 1 + t*z*G^2 + z*(G-1).
T(n, k) = 4^k*[x^k]hypergeometric([3/2, -n], [3], -x). - Peter Luschny, Feb 03 2015, based on an observation of Peter Bala in A254632.
T(n, k) = (-1)^k*Catalan(k+1)*Pochhammer(-n,k)/k!. - Peter Luschny, Feb 05 2015

Edited by N. J. A. Sloane at the suggestion of Andrew Plewe, Jun 16 2007

A247495 Generalized Motzkin numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k](exp(nx)* BesselI_{1}(2*x)/x), n>=0, k>=0.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 2, 4, 5, 3, 1, 0, 9, 14, 10, 4, 1, 5, 21, 42, 36, 17, 5, 1, 0, 51, 132, 137, 76, 26, 6, 1, 14, 127, 429, 543, 354, 140, 37, 7, 1, 0, 323, 1430, 2219, 1704, 777, 234, 50, 8, 1, 42, 835, 4862, 9285, 8421, 4425, 1514, 364, 65, 9, 1

Offset: 0

Peter Luschny, Dec 11 2014

This two-dimensional array of numbers can be seen as a generalization of the Motzkin numbers A001006 for two reasons: The case n=1 reduces to the Motzkin numbers and the columns are the values of the Motzkin polynomials M_{k}(x) = sum_{j=0..k} A097610(k,j)*x^j evaluated at the nonnegative integers.

			Square array starts:
[n\k][0][1] [2]  [3]   [4]   [5]    [6]     [7]      [8]
[0]   1, 0,  1,   0,    2,    0,     5,      0,      14, ...  A126120
[1]   1, 1,  2,   4,    9,   21,    51,    127,     323, ...  A001006
[2]   1, 2,  5,  14,   42,  132,   429,   1430,    4862, ...  A000108
[3]   1, 3, 10,  36,  137,  543,  2219,   9285,   39587, ...  A002212
[4]   1, 4, 17,  76,  354, 1704,  8421,  42508,  218318, ...  A005572
[5]   1, 5, 26, 140,  777, 4425, 25755, 152675,  919139, ...  A182401
[6]   1, 6, 37, 234, 1514, 9996, 67181, 458562, 3172478, ...  A025230
A000012,A001477,A002522,A079908, ...
.
Triangular array starts:
              1,
             0, 1,
           1, 1, 1,
          0, 2, 2, 1,
        2, 4, 5, 3, 1,
      0, 9, 14, 10, 4, 1,
   5, 21, 42, 36, 17, 5, 1,
0, 51, 132, 137, 76, 26, 6, 1.

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Main diagonal gives A247496.

Cf. A126120, A001006, A000108, A002212, A005572, A182401, A025230, A002522, A079908, A055151, A097610, A247497, A306684.

# RECURRENCE
T := proc(n,k) option remember; if k=0 then 1 elif k=1 then n else
(n*(2*k+1)*T(n,k-1)-(n-2)*(n+2)*(k-1)*T(n,k-2))/(k+2) fi end:
seq(print(seq(T(n,k),k=0..9)),n=0..6);
# OGF (row)
ogf := n -> (1-n*x-sqrt(((n-2)*x-1)*((n+2)*x-1)))/(2*x^2):
seq(print(seq(coeff(series(ogf(n),x,12),x,k),k=0..9)),n=0..6);
# EGF (row)
egf := n -> exp(n*x)*hypergeom([],[2],x^2):
seq(print(seq(k!*coeff(series(egf(n),x,k+2),x,k),k=0..9)),n=0..6);
# MOTZKIN polynomial (column)
A097610 := proc(n,k) if type(n-k,odd) then 0 else n!/(k!*((n-k)/2)!^2* ((n-k)/2+1)) fi end: M := (k,x) -> add(A097610(k,j)*x^j,j=0..k):
seq(print(seq(M(k,n),n=0..9)),k=0..6);
# OGF (column)
col := proc(n, len) local G; G := A247497_row(n); (-1)^(n+1)* add(G[k+1]/(x-1)^(k+1), k=0..n); seq(coeff(series(%, x, len+1),x,j), j=0..len) end: seq(print(col(n,8)), n=0..6); # Peter Luschny, Dec 14 2014

T[0, k_] := If[EvenQ[k], CatalanNumber[k/2], 0];
T[n_, k_] := n^k*Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4/n^2];
Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

def A247495(n,k):
    if n==0: return(k//2+1)*factorial(k)/factorial(k//2+1)^2 if is_even(k) else 0
    return n^k*hypergeometric([(1-k)/2,-k/2],[2],4/n^2).simplify()
for n in (0..7): print([A247495(n,k) for k in range(11)])

T(n,k) = (n*(2*k+1)*T(n,k-1)-(n-2)*(n+2)*(k-1)*T(n,k-2))/(k+2) for k>=2.
T(n,k) = Sum_{j=0..floor(k/2)} n^(k-2*j)*binomial(k,2*j)*binomial(2*j,j)/(j+1).
T(n,k) = n^k*hypergeom([(1-k)/2,-k/2], [2], 4/n^2) for n>0.
T(n,n) = A247496(n).
O.g.f. for row n: (1-n*x-sqrt(((n-2)*x-1)*((n+2)*x-1)))/(2*x^2).
O.g.f. for row n: R(x)/x where R(x) is series reversion of x/(1+n*x+x^2).
E.g.f. for row n: exp(n*x)*hypergeom([],[2],x^2).
O.g.f. for column k: the k-th column consists of the values of the k-th Motzkin polynomial M_{k}(x) evaluated at x = 0,1,2,...; M_{k}(x) = sum_{j=0..k} A097610(k,j)*x^j = sum_{j=0..k} (-1)^j*binomial(k,j)*A001006(j)*(x+1)^(k-j).
O.g.f. for column k: sum_{j=0..k} (-1)^(k+1)*A247497(k,j)/(x-1)^(j+1). - Peter Luschny, Dec 14 2014
O.g.f. for row n: 1/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017
T(n,k) is the coefficient of x^k in the expansion of 1/(k+1) * (1 + n*x + x^2)^(k+1). - Seiichi Manyama, May 07 2019

A304788 Expansion of e.g.f. exp(Sum_{k>=1} binomial(2k,k)x^k/(k + 1)!).

Original entry on oeis.org

1, 1, 3, 12, 59, 343, 2295, 17307, 144751, 1326377, 13189945, 141271298, 1619488645, 19766050827, 255693112641, 3492065507376, 50180426293255, 756444290843433, 11930511611596861, 196404976143077964, 3367697323914503113, 60029614473492823771, 1110430594720934758781

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

Exponential transform of A000108.

			E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 12*x^3/3! + 59*x^4/4! + 343*x^5/5! + 2295*x^6/6! + 17307*x^7/7! + ...

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A001517, A002212, A007317, A080893, A213507, A243953, A302197.

a:=series(exp(add(binomial(2*k,k)*x^k/(k+1)!,k=1..100)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Exp[Sum[CatalanNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] - BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[CatalanNumber[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} A000108(k)*x^k/k!).

E.g.f.: exp(exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)) - 1).

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).

Original entry on oeis.org

1, 1, 3, 19, 219, 3901, 95838, 3022909, 116798643, 5350403737, 283728025998, 17104314563843, 1155635807408096, 86513627563199279, 7109252862969177287, 636268582522962837475, 61610670571434193189443, 6418044336586421956746033, 715718717341021991299583730

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A002212, A307678, A349331, A349332, A349333, A349334, A349335.

Cf. A377098, A378325, A378327.

Table[Sum[Binomial[n-1, k-1]*Binomial[n*k, k]/((n-1)*k+1), {k, 0, n}], {n, 0, 20}]

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).

cp's OEIS Frontend

A371486 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^4.

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A384747 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,..,k}, and no nodes have the same weight as their parent node.

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A045445 Number of nonisomorphic systems of catafusenes for the unsymmetrical schemes (group C_s) with two appendages (see references for precise definition).

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A371483 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^3.

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A385563 Expansion of 1/((1-x) * (1-5*x))^(3/2).

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A002215 Number of polyhexes with n hexagons, having reflectional symmetry (see Harary and Read for precise definition).

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A108198 Triangle read by rows: T(n,k) = binomial(2k+2,k+1)*binomial(n,k)/(k+2) (0 <= k <= n).

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A247495 Generalized Motzkin numbers: Square array read by descending antidiagonals, T(n, k) = k!*[x^k](exp(n*x)* BesselI_{1}(2*x)/x), n>=0, k>=0.

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A247495 Generalized Motzkin numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k](exp(nx)* BesselI_{1}(2*x)/x), n>=0, k>=0.

A304788 Expansion of e.g.f. exp(Sum_{k>=1} binomial(2k,k)x^k/(k + 1)!).

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).