A068764
Generalized Catalan numbers 2*x*A(x)^2 -A(x) +1 -x =0.
Original entry on oeis.org
1, 1, 4, 18, 88, 456, 2464, 13736, 78432, 456416, 2697088, 16141120, 97632000, 595912960, 3665728512, 22703097472, 141448381952, 885934151168, 5575020435456, 35230798994432, 223485795258368, 1422572226146304, 9083682419818496, 58169612565614592, 373486362257899520, 2403850703479816192
Offset: 0
G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 456*x^5 + 2464*x^6 + 13736*x^7 + ...
-
Table[SeriesCoefficient[(1-Sqrt[1-8*x*(1-x)])/(4*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
Round@Table[4^(n-1) Hypergeometric2F1[(1-n)/2, 1-n/2, 2, 1/2] + KroneckerDelta[n]/Sqrt[2], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4^(n - 1) Hypergeometric2F1[ (1 - n)/2, (2 - n)/2, 2, 1/2]]; (* Michael Somos, Nov 08 2015 *)
-
a(n):=sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n); /* Vladimir Kruchinin, Aug 11 2010 */
-
{a(n) = my(A); if( n<1, n==0, n--; A = x * O(x^n); n! * simplify( polcoeff( exp(4*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* Michael Somos, Mar 31 2007 */
-
x='x+O('x^66); Vec((1-sqrt(1-8*x*(1-x)))/(4*x)) \\ Joerg Arndt, May 06 2013
A011117
Triangle of numbers S(x,y) = number of lattice paths from (0,0) to (x,y) that use step set { (0,1), (1,0), (2,0), (3,0), ....} and never pass below y = x.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 1, 3, 7, 11, 1, 4, 12, 28, 45, 1, 5, 18, 52, 121, 197, 1, 6, 25, 84, 237, 550, 903, 1, 7, 33, 125, 403, 1119, 2591, 4279, 1, 8, 42, 176, 630, 1976, 5424, 12536, 20793, 1, 9, 52, 238, 930, 3206, 9860, 26832, 61921, 103049, 1, 10, 63
Offset: 0
Robert Sulanke (sulanke(AT)diamond.idbsu.edu)
Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 2, 3]
[3] [1, 3, 7, 11]
[4] [1, 4, 12, 28, 45]
[5] [1, 5, 18, 52, 121, 197]
[6] [1, 6, 25, 84, 237, 550, 903]
[7] [1, 7, 33, 125, 403, 1119, 2591, 4279]
[8] [1, 8, 42, 176, 630, 1976, 5424, 12536, 20793]
[9] [1, 9, 52, 238, 930, 3206, 9860, 26832, 61921, 103049]
- E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, ECO method and hill-free generalized Motzkin paths, Séminaire Lotharingien de Combinatoire, B46b (2001), 14 pp.
- E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
-
f[ x_, y_ ] := f[ x, y ] = Module[ {return}, If[ x == 0, return = 1, If[ y == x-1, return = 0, return = f[ x, y-1 ] + Sum[ f[ k, y ], {k, 0, x-1} ] ] ]; return ]; Do[ Print[ Table[ f[ k, j ], {k, 0, j} ] ], {j, 10, 0, -1} ]
-
def A011117_row(n):
@cached_function
def prec(n, k):
if k==n: return 1
if k==0: return 0
return prec(n-1,k-1)+sum(prec(n,k+i-1) for i in (2..n-k+1))
return [prec(n, n-k) for k in (0..n-1)]
for n in (1..9): print(A011117_row(n)) # Peter Luschny, Mar 16 2016
A034015
Small 3-Schroeder numbers: a(n) = A027307(n+1)/2.
Original entry on oeis.org
1, 5, 33, 249, 2033, 17485, 156033, 1431281, 13412193, 127840085, 1235575201, 12080678505, 119276490193, 1187542872989, 11909326179841, 120191310803937, 1219780566014657, 12440630635406245, 127446349676475425, 1310820823328281561, 13530833791486094769
Offset: 0
- Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.
- Alois P. Heinz, Table of n, a(n) for n = 0..500
- J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
- Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.4.
Part of a family indexed by m: m=2 (
A001003), m=3 is this sequence, m=4 is
A243675, ....
-
a:= proc(n) option remember; `if`(n<2, 4*n+1,
((110*n^3+66*n^2-17*n-9) *a(n-1)
+(n-1)*(2*n-1)*(5*n+3) *a(n-2)) /
((2*n+3)*(5*n-2)*(n+1)))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Jun 22 2014
-
a[n_] := If[n<0, 0, Sum[2^i*Binomial[2*n+2, i]*Binomial[n+1, i+1]/(n+1), {i, 0, n}]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 13 2014, after PARI *)
a[n_] := Hypergeometric2F1[-n, -2 (n + 1), 2, 2];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Nov 08 2021 *)
-
a(n)=if(n<0,0,sum(i=0,n,2^i*binomial(2*n+2,i)*binomial(n+1,i+1))/(n+1))
A082298
Expansion of (1-3*x-sqrt(9*x^2-10*x+1))/(2*x).
Original entry on oeis.org
1, 4, 20, 116, 740, 5028, 35700, 261780, 1967300, 15072836, 117297620, 924612532, 7367204260, 59240277988, 480118631220, 3917880562644, 32163325863300, 265446382860420, 2201136740855700, 18329850024033012, 153225552507991140
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
- Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
- Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
- Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
- Zhi Chen and Hao Pan, Identities involving weighted Catalan, Schroder and Motzkin paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=4, b=1.
- Curtis Coker, Enumerating a class of lattice paths, Disc. Math. (2003) Vol. 271, Issues 1-3, 13-28.
- Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
- John Machacek, Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats, arXiv:2105.02417 [math.CO], 2021. See Thm 4.4, G(x,F^1)
- Greg Morrow, Some probability distributions and integer sequences related to rook paths, Univ. Colorado Springs (2024). See pp. 4, 22
-
Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-3*x-Sqrt(9*x^2-10*x+1))/(2*x))); // G. C. Greubel, Feb 10 2018
-
A082298_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 4*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od;convert(a,list)end: A082298_list(20); # Peter Luschny, May 19 2011
a := n -> `if`(n=0, 1, 4*hypergeom([1 - n, -n], [2], 4)):
seq(simplify(a(n)), n=0..20); # Peter Luschny, May 22 2017
-
gf[x_] = (1 - 3*x - Sqrt[(9*x^2 - 10*x + 1)])/(2*x); CoefficientList[Series[gf[x], {x, 0, 20}], x] (* Jean-François Alcover, Jun 01 2011 *)
-
a(n)=if(n<1,1,sum(k=0,n,4^k*binomial(n,k)*binomial(n,k-1))/n)
A111785
T(n,k) are coefficients used for power series inversion (sometimes called reversion), n >= 0, k = 1..A000041(n), read by rows.
Original entry on oeis.org
1, -1, -1, 2, -1, 5, -5, -1, 6, 3, -21, 14, -1, 7, 7, -28, -28, 84, -42, -1, 8, 8, 4, -36, -72, -12, 120, 180, -330, 132, -1, 9, 9, 9, -45, -90, -45, -45, 165, 495, 165, -495, -990, 1287, -429, -1, 10, 10, 10, 5, -55, -110, -110, -55, -55, 220, 660, 330, 660, 55, -715, -2860, -1430, 2002, 5005, -5005, 1430, -1, 11, 11
Offset: 0
[ +1];
[ -1];
[ -1, 2];
[ -1, 5, -5];
[ -1, 6, 3, -21, 14];
[ -1, 7, 7, -28, -28, 84, -42];
[ -1, 8, 8, 4, -36, -72, -12, 120, 180, -330, 132]; ...
The seventh row, [ -1, 8, 8, 4, -36, -72, -12, 120, 180, -330, 132], stands for the row polynomial P(6) with monomials in lexicographically ascending order P(6) = -1*g[0]^5*g[6] + 8*g[0]^4*g[1]*g[5] + 8*g[0]^4*g[2]*g[4] + 4*g[0]^4*g[3]^2 - 36*g[0]^3*g[1]^2*g[4] - 72*g[0]^3*g[1]*g[2]*g[3] - 12*g[0]^3*g[2]^3 + 120*g[0]^2*g[1]^3*g[3] + 180*g[0]^2*g[1]^2*g[2]^2 - 330*g[0]*g[1]^4*g[2] + 132*g[1]^6 = (1/7!)*(differentiate 1/G(x)^7 six times and evaluate at x = 0). This gives the coefficient of y^7 of F^{(-1)}(y).
- J. Riordan, Combinatorial Identities, Wiley, 1968, p. 150, Table 4.1 (unsigned).
- Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16, 3.6.25.
- Bartomeu Fiol and Alan Rios Fukelman, On the planar free energy of matrix models, arXiv:2111.14783 [hep-th], 2021. See also J. High Energy Phys. (2022) Iss. 2.
- Wolfdieter Lang, First 10 rows and a formula.
- Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
- Eric Weisstein's World of Mathematics, Series Reversion
Row sums give (-1)^n. Unsigned row sums are
A001003(n) (little Schroeder numbers). Inversion triangle with leading quadratic term:
A276738. Conjectured simplification:
A283298.
-
(* Graded Colex Ordering: by length, then reverse lexicographic by digit *)
ClearAll[P, L, T, c, g]
P[0] := 1
P[n_] := -Total[
Multinomial @@ # c[Total@# - 1] Times @@
Power[g[#] & /@ Range[0, n - 1], #] & /@
Table[ Count[p, i], {p, Drop[IntegerPartitions[n + 1], 1]}, {i,
n}]]
L[n_] := Join @@ GatherBy[IntegerPartitions[n], Length]
T[1] := {1}
T[n_] := Coefficient[ Do[g[i] = P[i], {i, 0, n - 1}];
P[n - 1], #] & /@ (Times @@@ Map[c, L[n - 1], {2}])
Array[T, 9] // Flatten (* Bradley Klee and Michael Somos, Apr 14 2017 *)
-
sv(n)={eval(Str("'s",n))}
Trm(q,v)={my(S=Set(v)); for(i=1, #S, my(x=S[i], c=#select(y->y==x, v)); q=polcoef(q, c, sv(x))); q}
Q(n)={polcoef(serreverse(x + x*sum(k=1, n, x^k*sv(k), O(x*x^n)))/x, n)}
row(n)={my(q=Q(n)); [Trm(q,Vec(v)) | v<-partitions(n)]} \\ Andrew Howroyd, Feb 01 2022
-
C(v)={my(n=vecsum(v), S=Set(v)); (-1)^#v*(n+#v)!/(n+1)!/prod(i=1, #S, my(x=S[i], c=#select(y->y==x, v)); c!)}
row(n)=[C(Vec(p)) | p<-partitions(n)]
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022
-
def A111785_list(dim): # returns the first dim rows
C = [[0 for k in range(m+1)] for m in range(dim+1)]
C[0][0] = 1; F = [1]; i = 1
X = lambda n: 1 if n == 1 else var('x'+str(n))
while i <= dim: F.append(F[i-1]*X(i)); i += 1
for m in (1..dim):
C[m][m] = -C[m-1][m-1]/F[1]
for k in range(m-1, 0, -1):
C[m][k] = -(C[m-1][k-1]+sum(F[i]*C[m][k+i-1] for i in (2..m-k+1)))/F[1]
P = [expand((-1)^m*C[m][1]) for m in (1..dim)]
R = PolynomialRing(ZZ, [X(i) for i in (2..dim)], order='lex')
return [R(p).coefficients()[::-1] for p in P]
A111785_list(8) # Peter Luschny, Apr 14 2017
A181289
Triangle read by rows: T(n,k) is the number of 2-compositions of n having length k (0 <= k <= n).
Original entry on oeis.org
1, 0, 2, 0, 3, 4, 0, 4, 12, 8, 0, 5, 25, 36, 16, 0, 6, 44, 102, 96, 32, 0, 7, 70, 231, 344, 240, 64, 0, 8, 104, 456, 952, 1040, 576, 128, 0, 9, 147, 819, 2241, 3400, 2928, 1344, 256, 0, 10, 200, 1372, 4712, 9290, 11040, 7840, 3072, 512, 0, 11, 264, 2178, 9108, 22363
Offset: 0
Triangle starts:
1;
0, 2;
0, 3, 4;
0, 4, 12, 8;
0, 5, 25, 36, 16;
0, 6, 44, 102, 96, 32;
0, 7, 70, 231, 344, 240, 64;
0, 8, 104, 456, 952, 1040, 576, 128;
0, 9, 147, 819, 2241, 3400, 2928, 1344, 256;
0, 10, 200, 1372, 4712, 9290, 11040, 7840, 3072, 512;
0, 11, 264, 2178, 9108, 22363, 34332, 33488, 20224, 6912, 1024;
- John Tyler Rascoe, Rows n = 0..100, flattened
- G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
- Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (11).
-
T := proc (n, k) if k <= n then sum((-1)^j*2^(k-j)*binomial(k, j)*binomial(n+k-j-1, 2*k-1), j = 0 .. k) else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
# Uses function PMatrix from A357368.
PMatrix(10, n -> n + 1); # Peter Luschny, Oct 19 2022
-
Table[Sum[(-1)^j*2^(k - j) Binomial[k, j] Binomial[n + k - j - 1, 2 k - 1], {j, 0, k}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 11 2015 *)
-
T_xt(max_row) = {my(N=max_row+1, x='x+O('x^N), h=(1-x)^2/((1-x)^2 - t*x*(2-x))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_xt(10) \\ John Tyler Rascoe, Apr 05 2025
A317875
Number of achiral free pure multifunctions with n unlabeled leaves.
Original entry on oeis.org
1, 1, 3, 9, 30, 102, 369, 1362, 5181, 20064, 79035, 315366, 1272789, 5185080, 21296196, 88083993, 366584253, 1533953100, 6449904138, 27238006971, 115475933202, 491293053093, 2096930378415, 8976370298886, 38528771056425, 165784567505325
Offset: 1
The first 4 terms count the following multifunctions.
o,
o[o],
o[o,o], o[o[o]], o[o][o],
o[o,o,o], o[o[o][o]], o[o[o[o]]], o[o[o,o]], o[o][o,o], o[o][o[o]], o[o][o][o], o[o,o][o], o[o[o]][o].
Cf.
A001003,
A001678,
A002033,
A003238,
A052893,
A053492,
A067824,
A167865,
A214577,
A277996,
A280000,
A317853.
-
a[n_]:=If[n==1,1,Sum[a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,12]
-
seq(n)={my(p=O(x)); for(n=1, n, p = x + p*(sum(k=1, n-1, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018
-
seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-1, v[i]*sumdiv(n-i, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018
A078009
a(0)=1, for n>=1 a(n) = Sum_{k=0..n} 5^k*N(n,k) where N(n,k) = C(n,k)*C(n,k+1)/n are the Narayana numbers (A001263).
Original entry on oeis.org
1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
- Xiaomei Chen, Yuan Xiang, Counting generalized Schröder paths, arXiv:2009.04900 [math.CO], 2020.
-
R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+4*x - Sqrt(16*x^2-12*x+1))/(10*x) )); // G. C. Greubel, Jun 28 2019
-
[1] cat [&+[5^k*Binomial(n,k)*Binomial(n,k+1)/n:k in [0..n]]:n in [1..20]]; // Marius A. Burtea, Jan 21 2020
-
A078009_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+5*add(a[j]*a[w-j-1],j=1..w-1) od;
convert(a, list) end: A078009_list(20); # Peter Luschny, May 19 2011
-
Table[SeriesCoefficient[(1+4*x-Sqrt[16*x^2-12*x+1])/(10*x),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 5];
Table[a[n], {n, 0, 30}] (* Peter Luschny, Mar 19 2018 *)
-
a(n)=sum(k=0,n,5^k/n*binomial(n,k)*binomial(n,k+1))
-
a=((1+4*x -sqrt(16*x^2-12*x+1))/(10*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 28 2019
A364748
G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)).
Original entry on oeis.org
1, 1, 6, 47, 424, 4159, 43097, 464197, 5145475, 58313310, 672598269, 7869856070, 93183973405, 1114471042413, 13443614108307, 163372291277764, 1998239045199623, 24580340878055298, 303893356012560280, 3774099648814193998, 47061518776483143441
Offset: 0
-
a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(5*n-4*k, n-1-k))/n);
-
a(n, r=1, s=1, t=5, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 05 2024
A006241
Number of minimal plane trees with n terminal nodes.
Original entry on oeis.org
1, 1, 1, 2, 1, 3, 1, 6, 2, 3, 1, 20, 1, 3, 3, 54, 1, 34, 1, 44, 3, 3, 1, 764, 2, 3, 10, 140, 1, 283, 1, 4470, 3, 3, 3, 10416, 1, 3, 3, 10820, 1, 2227, 1, 2060, 62, 3, 1, 958476, 2, 250, 3, 8204, 1, 59154, 3, 316004, 3, 3, 1, 3457904, 1, 3, 158, 30229110, 3
Offset: 1
The a(12)=20 same-trees with all leaves greater than 1 are:
12, (3333), (222222), ((33)(33)), ((33)(222)), ((33)6), ((222)(33)), ((222)(222)), ((222)6), (6(33)), (6(222)), (66), ((22)(22)(22)), ((22)(22)4), ((22)4(22)), ((22)44), (4(22)(22)), (4(22)4), (44(22)), (444). - _Gus Wiseman_, Jan 15 2017
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
a:= proc(n) option remember; `if`(n=1, 1, add(
a(n/d)^d, d=numtheory[divisors](n) minus {1}))
end:
seq(a(n), n=1..70); # Alois P. Heinz, Feb 21 2017
-
Array[If[#1===1,1,Sum[#0[#1/d]^d,{d,Rest[Divisors[#1]]}]]&,200] (* Gus Wiseman, Jan 15 2017 *)
Comments