A007317 -id:A007317 - cp's OEIS frontend

A202060 Number of ascent sequences avoiding the pattern 110.

1, 1, 2, 5, 14, 43, 143, 510, 1936, 7774, 32848, 145398, 671641, 3227218, 16084747, 82955090, 441773793, 2424845273, 13695855478, 79485625385, 473393639992, 2889930405750, 18064609329598, 115513453404597, 754956282308784, 5039064184597772, 34323984497482559

Offset: 0

N. J. A. Sloane, Dec 10 2011

nonn

It appears that no formula or g.f. is known.

Andrew Conway and Miles Conway, Table of n, a(n) for n = 0..43
Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641, 2011

Total number of ascent sequences is given by A022493. Number of ascent sequences avoiding 001 (and others) is A000079; 102 is A007051; 101 is A000108; 000 is A202058; 100 is A202059; 110 is A202060; 120 is A202061; 201 is A202062; 210 is A108304; 0123 is A080937; 0021 is A007317.

a(0) and a(15)-a(17) from Alois P. Heinz, Jan 07 2014

More terms from Anthony Guttmann, Nov 04 2021

A202061 Number of ascent sequences avoiding the pattern 120.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 133, 442, 1535, 5546, 20754, 80113, 317875, 1292648, 5374073, 22794182, 98462847, 432498659, 1929221610, 8728815103, 40017844229, 185727603829, 871897549029, 4137132922197, 19828476952117, 95934298966615, 468291607852143, 2305162065138433

Offset: 0

N. J. A. Sloane, Dec 10 2011

nonn

It appears that no formula or g.f. is known.

Liang Chengwei, Shi Lecun and Cai Zhongyu, Table of n, a(n) for n = 0..500 (terms 0..74 from Andrew Conway and Miles Conway)
Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
Paul Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.

Total number of ascent sequences is given by A022493. Number of ascent sequences avoiding 001 (and others) is A000079; 102 is A007051; 101 is A000108; 000 is A202058; 100 is A202059; 110 is A202060; 120 is A202061; 201 is A202062; 210 is A108304; 0123 is A080937; 0021 is A007317.

More terms from Anthony Guttmann, Nov 04 2021

A270447 Binomial transform(2) of Catalan numbers.

Original entry on oeis.org

1, 3, 11, 43, 174, 721, 3044, 13059, 56837, 250690, 1119612, 5059561, 23119628, 106753404, 497762380, 2342096579, 11113027686, 53138757319, 255892224332, 1240217043450, 6046131132030, 29631889507380, 145923474439800, 721733515299225, 3583733352377724

Offset: 0

Vladimir Kruchinin, Mar 17 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A007317, A092392.

Table[Sum[Binomial[2*k,k]/(k+1) * Binomial[2*n-k,n], {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, Mar 17 2016 *)
a[n_] := ((2 n + 1) Binomial[2 n, n] (1 - Hypergeometric2F1[-1/2, -n - 1, -2 n - 1, 4]))/(2 (n + 1));
Table[a[n], {n, 0, 24}] (* Peter Luschny, May 30 2022 *)

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

a(n) = sum(i=0, n, (binomial(2*i, i)*binomial(2*n-i, n))/(i+1)); \\ Altug Alkan, Mar 17 2016

a(n) = Sum_{k=0..n} (T(n,k)*C(k)), where C(k) is Catalan numbers (A000108), T(n,k) - triangle of A092392.
a(n) = Sum_{k=0..n} ((binomial(2*k,k)/(k+1)*binomial(2*n-k,n))).
G.f.: C(C(x))*(1-C(x))^2/(((1-C(x))^2)-x)/x, where C(x)=(1-sqrt(1-4*x))/2.
Recurrence: 3*(n-1)*n*(n+1)*(2*n - 3)*a(n) = 16*(n-1)*n*(5*n^2 - 10*n + 3)*a(n-1) - 16*(n-1)*(2*n - 1)*(11*n^2 - 33*n + 24)*a(n-2) + 8*(2*n - 3)*(2*n - 1)*(4*n - 9)*(4*n - 7)*a(n-3). - Vaclav Kotesovec, Mar 17 2016
a(n) ~ 2^(4*n + 1/2) / (sqrt(Pi) * 3^(n - 1/2) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2016
a(n) = [x^n] (1 - sqrt(1 - 4*x))/(2*x*(1 - x)^(n+1)). - Ilya Gutkovskiy, Nov 01 2017

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).

A340968 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^jbinomial(n,j)Catalan(j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 15, 1, 1, 5, 25, 71, 51, 1, 1, 6, 41, 199, 441, 188, 1, 1, 7, 61, 429, 1795, 2955, 731, 1, 1, 8, 85, 791, 5073, 17422, 20805, 2950, 1, 1, 9, 113, 1315, 11571, 64469, 177463, 151695, 12235, 1

Offset: 0

Seiichi Manyama, Jan 31 2021

			Square array begins:
  1,   1,    1,     1,     1,      1, ...
  1,   2,    3,     4,     5,      6, ...
  1,   5,   13,    25,    41,     61, ...
  1,  15,   71,   199,   429,    791, ...
  1,  51,  441,  1795,  5073,  11571, ...
  1, 188, 2955, 17422, 64469, 181776, ...

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

Columns k=0..4 give A000012, A007317(n+1), A162326(n+1), A337167, A386387.
Main diagonal gives A338979.
Cf. A000108, A290605, A340970.

T_row := n -> k -> hypergeom([1/2, -n], [2], -4*k): for n from 0 to 6 do seq(simplify(T_row(n)(k)), k = 0..6) od; # Peter Luschny, Aug 27 2025

T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * CatalanNumber[j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 01 2021 *)
A340968[n_, k_] := Hypergeometric2F1[1/2, -n, 2, -4*k]; Table[A340968[n, k], {n, 0, 6}, {k, 0, 7}] (* row-wise *) (* Peter Luschny, Aug 27 2025 *)

T(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*j)!/(j!*(j+1)!));

T(n, k) = 1+k*sum(j=0, n-1, T(j, k)*T(n-1-j, k));

G.f. A_k(x) of column k satisfies A_k(x) = 1/(1 - x) + k*x*A_k(x)^2.
A_k(x) = 2/( 1 - x + sqrt((1 - x) * (1 - (4*k+1)*x)) ).
T(n,k) = 1 + k * Sum_{j=0..n-1} T(j,k) * T(n-1-j,k).
(n+1) * T(n,k) = 2 * ((2*k+1) * n - k) * T(n-1,k) - (4*k+1) * (n-1) * T(n-2,k) for n > 1.
E.g.f. of column k: exp((2*k+1)*x) * (BesselI(0,2*k*x) - BesselI(1,2*k*x)). - Ilya Gutkovskiy, Feb 01 2021
T_row(n) = k -> hypergeom([1/2, -n], [2], -4*k). - Peter Luschny, Aug 27 2025

A348858 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(3*x))).

Original entry on oeis.org

1, 2, 9, 103, 3101, 261192, 64285189, 47059492688, 103060910397021, 676492249628112382, 13317427360663454672669, 786420726604930579016189223, 139314431838014895142151741877241, 74037818920801629179455290512454633872, 118040419689979917511971388549088825283510249

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A007317, A015084, A348857, A348859.

nmax = 14; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[3 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[3^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]

a(n) = 1 + Sum_{k=0..n-1} 3^k * a(k) * a(n-k-1).

a(n) ~ c * 3^(n*(n-1)/2), where c = 4.508135635010167805309616576501854361005320931661829410476785686203732753... - Vaclav Kotesovec, Nov 02 2021

A360322 a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, -4, 10, -30, 102, -376, 1462, -5900, 24470, -103644, 446382, -1948854, 8605290, -38362200, 172423770, -780496110, 3554991270, -16281079900, 74927379550, -346328465930, 1607078948690, -7483861047480, 34963419415650, -163825013554400, 769694347677002

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360317, A360318, A360319, A360321.

Cf. A007317.

a(n) = sum(k=0, n, (-5)^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));

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

G.f.: sqrt( (1+5*x)/(1+x) ).
n*a(n) = 2*(-3*n+4)*a(n-1) - 5*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (-1/5)^i * a(j) * a(i-j) = (1/5)^n.
a(n) = 2 * (-1)^(n+1) * A007317(n) for n > 0.
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (-1/4)^n * Sum_{k=0..n} 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = (-1)^n * Sum_{k=0..n} binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A367692 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x^4))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 13, 20, 29, 43, 66, 102, 155, 233, 352, 536, 817, 1240, 1878, 2848, 4327, 6576, 9984, 15150, 22995, 34919, 53029, 80513, 122224, 185556, 281736, 427776, 649481, 986054, 1497069, 2272976, 3451038, 5239607, 7955067, 12077876, 18337503

Offset: 0

Seiichi Manyama, Nov 27 2023

nonn

Cf. A007317, A351972, A367691.

Cf. A367660.

a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, (i-1)\4, v[j+1]*v[i-4*j])); v;

from functools import lru_cache
@lru_cache(maxsize=None)
def A367692(n): return 1+sum(A367692(k)*A367692(n-1-(k<<2)) for k in range(n+3>>2)) # Chai Wah Wu, Nov 30 2023

a(n) = 1 + Sum_{k=0..floor((n-1)/4)} a(k) * a(n-1-4*k).

A369616 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^2 + x) ).

Original entry on oeis.org

1, 3, 12, 58, 314, 1824, 11107, 69955, 451918, 2977834, 19936332, 135225006, 927267595, 6417580459, 44770275705, 314489676679, 2222549047262, 15791353483602, 112734135824404, 808247711066688, 5817056710700424, 42012120642574732, 304384379305912686

Offset: 0

Seiichi Manyama, Jan 27 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A007317, A369617, A370844.

Cf. A006013, A274734.

A369616 := proc(n)
    add(binomial(n+1,k) * binomial(3*n-3*k+1,n-k),k=0..n) ;
    %/(n+1) ;
end proc;
seq(A369616(n),n=0..70) ; # R. J. Mathar, Jan 28 2024

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

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

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(3*n-3*k+1,n-k).

D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) +3*(-13*n^2+1)*a(n-1) +33*(2*n-1)*(n-1)*a(n-2) -31*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jan 28 2024

A369617 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^3 + x) ).

Original entry on oeis.org

1, 4, 22, 146, 1079, 8525, 70468, 601816, 5268241, 47019566, 426250277, 3914020148, 36328457669, 340278596273, 3212416054283, 30534649412247, 291981031204917, 2806832429353512, 27109863184695640, 262951127248539898, 2560229132085602215

Offset: 0

Seiichi Manyama, Jan 27 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..984
Index entries for reversions of series

Cf. A007317, A369616, A370844.

Cf. A006632, A274735.

A369617 := proc(n)
    add(binomial(n+1,k) * binomial(4*n-4*k+2,n-k),k=0..n) ;
    %/(n+1) ;
end proc;
seq(A369617(n),n=0..70) ; # R. J. Mathar, Jan 28 2024

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

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

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(4*n-4*k+2,n-k).

D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-91*n^3 -32*n^2 +n+2)*a(n-1) +2*(n-1)*(465*n^2 -337*n+86)*a(n-2) -4*(n-1)*(n-2) *(219*n-187)*a(n-3) +283*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 28 2024

cp's OEIS Frontend

A202060 Number of ascent sequences avoiding the pattern 110.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A202061 Number of ascent sequences avoiding the pattern 120.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A270447 Binomial transform(2) of Catalan numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A340968 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j*binomial(n,j)*Catalan(j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A348858 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(3*x))).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A360322 a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A367692 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x^4))).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

A369616 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^2 + x) ).

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

A340968 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^jbinomial(n,j)Catalan(j).