A186997 -id:A186997 - cp's OEIS frontend

A192479 a(n) = 2^n*C(n-1) - A186997(n-1), where C(n) are the Catalan numbers (A000108).

1, 3, 12, 61, 344, 2074, 13080, 85229, 569264, 3876766, 26817304, 187908802, 1330934032, 9513485076, 68539442800, 497178707325, 3628198048352, 26617955242806, 196205766112536, 1452410901340598, 10792613273706320

Offset: 1

Volkan Yildiz, Jul 01 2011

nonn

a(n) is the number of rows with the value true in the truth tables of all bracketed formulas with n distinct propositions connected by the binary connective of implication.

P. J. Cameron and V. Yildiz, Counting false entries in truth tables of bracketed formulas connected by implication, arXiv:1106.4443 [math.CO], 2011.

Cf. A186997.

C := proc(n) binomial(2*n,n)/(n+1) ;end proc:
Yildf := proc(n) option remember; if n<=1 then 1; else add( (2^i*C(i-1)-procname(i))*procname(n-i),i=1..n-1) ; end if; end proc:
A192479 := proc(n) 2^n*C(n-1)-Yildf(n) ; end proc:
seq(A192479(n),n=1..30) ; # R. J. Mathar, Jul 13 2011

a[1] = 1; a[n_] := 2^n*CatalanNumber[n-1] - Sum[Binomial[k, n-k-1]*Binomial[n+2*k-1, n+k-1]/(n+k), {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2015 *)

a(n) = 2^n*C(n) - f(n), with f(n) = Sum_{i=1..n-1} (2^i*C(i)-f(i))*f(n-i), starting f(0)=f(1)=1, where C(i) = A000108(i-1).
G.f.: 1 - 1/A186997(x). - Vladimir Kruchinin, Feb 17 2013
a(n+1) = Sum_{k=1..n+1} (binomial(k,n-k+1)*binomial(n+2*k-1,k))/(n+k), a(1)=1. - Vladimir Kruchinin, May 15 2014

A364475 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^3.

Original entry on oeis.org

1, 1, 4, 18, 94, 529, 3135, 19270, 121732, 785496, 5155167, 34304706, 230923653, 1569684910, 10759159000, 74281473504, 516089542684, 3605685460750, 25316226436086, 178538289189108, 1264131169628799, 8982889404251721, 64041351551534215

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Column k=1 of A378323.

Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364478.

A364475 := proc(n)
    add( binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k)/(2*n-2*k+1),k=0..n/2) ;
end proc:
seq(A364475(n),n=0..80); # R. J. Mathar, Jul 27 2023

a(n) = sum(k=0, n\2, binomial(3*n-3*k, k)*binomial(3*n-4*k, n-2*k)/(2*n-2*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k) / (2*n-2*k+1).

D-finite with recurrence 2*n*(2*n+1)*a(n) -(5*n+1)*(3*n-2)*a(n-1) +4*(-25*n^2+75*n-59) *a(n-2) +9*(-15*n^2+69*n-80)*a(n-3) -6*(3*n-8)*(3*n-10) *a(n-4)=0. - R. J. Mathar, Jul 27 2023

A218045 Number of truth tables of bracketed formulas (case 3).

Original entry on oeis.org

0, 0, 1, 2, 9, 46, 262, 1588, 10053, 65686, 439658, 2999116, 20774154, 145726348, 1033125004, 7390626280, 53281906861, 386732675046, 2823690230850, 20725376703324, 152833785130398, 1131770853856100, 8412813651862868

Offset: 0

N. J. A. Sloane, Oct 23 2012

nonn

Equals the self-convolution of A186997 (up to offset). - Paul D. Hanna, Jul 03 2023

			G.f. A(x) = x^2 + 2*x^3 + 9*x^4 + 46*x^5 + 262*x^6 + 1588*x^7 + 10053*x^8 + 65686*x^9 + 439658*x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv:1205.5595 [math.CO], 2012.

Cf. A186997, A218182, A288470.

CoefficientList[Series[(2+2*Sqrt[1-8*x]-(1+Sqrt[1-8*x])*Sqrt[2+2*Sqrt[1-8*x]+8*x])/8, {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 19 2014 after Yildiz *)
Flatten[{0,0,Table[Sum[(Sum[Binomial[k,2*k+i+2-n]*Binomial[k+i-1,i],{i,0,n-k-1}]*Binomial[2*n-2,k])/(n-1),{k,0,n-1}],{n,2,20}]}] (* Vaclav Kotesovec, Nov 19 2014 after Vladimir Kruchinin *)

a(n):=sum((sum(binomial(k,2*k+i-n)*binomial(k+i-1,i),i,0,n-k+1))*binomial(2*n+2,k),k,0,n+1)/(n+1); /* Vladimir Kruchinin, Nov 19 2014  */

x='x+O('x^50); concat([0,0], Vec((2+2*sqrt(1-8*x)-(1+sqrt(1-8*x))*sqrt(2 + 2*sqrt(1-8*x)+8*x))/8)) \\ G. C. Greubel, Apr 01 2017

Yildiz gives a g.f.: (2+2*sqrt(1-8*x)-(1+sqrt(1-8*x))*sqrt(2+2*sqrt(1-8*x)+8*x))/8.
a(n+1) = (Sum_{k = 0..n} (Sum_{i=0..n-k} (binomial(k, 2*k+i+1-n)*binomial(k+i-1, i)))*binomial(2*n,k))/n. - Vladimir Kruchinin, Nov 19 2014
G.f. G(x) = A(x)/x satisfies G(x) = x*((G(x)*(G(x)+1))/(1-G(x))+1)^2. - Vladimir Kruchinin, Nov 19 2014
a(n) ~ (2*sqrt(3)-3) * 2^(3*n-3) / (3 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 19 2014
From Paul D. Hanna, Jul 03 2023: (Start)
G.f. A(x) = Series_Reversion( x*(1 + sqrt(1 - 4*x - 4*x^2)) / 2 )^2.
G.f. A(x) = exp( Sum_{n>=1} A288470(n) * x^n/n ), where A288470(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n,2*k). (End)

A364474 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x).

Original entry on oeis.org

1, 1, 4, 16, 77, 403, 2228, 12800, 75653, 457022, 2809266, 17514200, 110480475, 703850686, 4522217364, 29268545416, 190645760149, 1248817411471, 8221323983431, 54365667330636, 360954069730636, 2405225494066647, 16080210766344354, 107828663888705292

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1175
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 26.

Cf. A002293, A104979, A186997, A255673, A361245, A364475, A364478.

A364474 := proc(n)
    add( binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k)/(2*n-4*k+1),k=0..n/2) ;
end proc:
seq(A364474(n),n=0..80); # R. J. Mathar, Jul 27 2023

Table[Sum[Binomial[3*n - 5*k, k]*Binomial[3*n - 6*k, n - 2*k]/(2*n - 4*k + 1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Wesley Ivan Hurt, May 25 2024 *)

a(n) = sum(k=0, n\2, binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k) / (2*n-4*k+1).

D-finite with recurrence 2*n*(2*n+1)*(3*n-7)*a(n) -3*(3*n-1)*(3*n-7)*(3*n-2) *a(n-1) -2*(n-3)*(18*n^2-33*n+4) *a(n-2) +2*(18*n^3-141*n^2+287*n-64) *a(n-4) -2*(n-4)*(3*n-1)*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 27 2023

A364478 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^8.

Original entry on oeis.org

1, 1, 4, 23, 154, 1124, 8675, 69626, 575243, 4859778, 41789764, 364565277, 3218581695, 28702642553, 258172627259, 2339496034381, 21337716782873, 195726876816623, 1804472496834650, 16711389876481027, 155395461519245354, 1450298253483719944

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475.

a(n) = sum(k=0, n\2, binomial(3*n+2*k, k)*binomial(3*n+k, n-2*k)/(2*n+3*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+2*k,k) * binomial(3*n+k,n-2*k) / (2*n+3*k+1).

A378290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n+2r+k,r) binomial(r,n-r)/(n+2*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 19, 0, 1, 4, 15, 46, 104, 0, 1, 5, 22, 82, 262, 614, 0, 1, 6, 30, 128, 486, 1588, 3816, 0, 1, 7, 39, 185, 789, 3027, 10053, 24595, 0, 1, 8, 49, 254, 1185, 5052, 19543, 65686, 162896, 0, 1, 9, 60, 336, 1689, 7801, 33290, 129606, 439658, 1101922, 0

Offset: 0

Seiichi Manyama, Nov 21 2024

			Square array begins:
  1,    1,     1,     1,     1,     1,     1, ...
  0,    1,     2,     3,     4,     5,     6, ...
  0,    4,     9,    15,    22,    30,    39, ...
  0,   19,    46,    82,   128,   185,   254, ...
  0,  104,   262,   486,   789,  1185,  1689, ...
  0,  614,  1588,  3027,  5052,  7801, 11430, ...
  0, 3816, 10053, 19543, 33290, 52490, 78552, ...

Columns k=0..2 give A000007, A186997, A218045(n+2).

Cf. A009766, A026300, A378289, A378291, A378292.

T(n, k, t=3, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + x * A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A186997.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-2,k+3) for n > 1.

A192481 a(n) = Sum_{i=1..n-1} (2^iC(i)-a(i)) (2^(n-i)*C(n-i)-a(n-i)), a(1)=1, where C(i)=A000108(i-1) are Catalan numbers.

Original entry on oeis.org

1, 1, 6, 29, 162, 978, 6156, 40061, 267338, 1819238, 12576692, 88079378, 623581332, 4455663876, 32090099352, 232711721757, 1697799727066, 12452943237342, 91774314536100, 679234371006982, 5046438870909244, 37623611703611452, 281391143518722728

Offset: 1

Volkan Yildiz, Jul 01 2011

nonn

a(n) is the number of rows with the value false in the truth tables of all bracketed m-implication, case (i), with n distinct variables.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Volkan Yildiz, Counting false entries in truth tables of bracketed formulas connected by m-implication, arXiv:1203.4645 [math.CO], 2012.
Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv:1205.5595 [math.CO], 2012.

Cf. A000108, A186997, A192479.

C := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A192481 := proc(n) option remember; if n<=1 then n; else add( (2^i*C(i-1)-procname(i))*(2^(n-i)*C(n-i-1)-procname(n-i)), i=1..n-1) ; end if; end proc:

CoefficientList[Series[(2 - Sqrt[1 - 8*x] - Sqrt[3 - 4*x - 2*Sqrt[1 - 8*x]])/2, {x,0,50}], x] (* G. C. Greubel, Feb 12 2017 *)

x='x+O('x^50); Vec((2-sqrt(1-8*x)-sqrt(3-4*x-2*sqrt(1-8*x)))/2) \\ G. C. Greubel, Feb 12 2017

G.f.: (2 - sqrt(1-8*x) - sqrt(3 - 4*x - 2*sqrt(1-8*x)))/2.
For large n, a(n) is asymptotically (1-2/sqrt 10) * 2^(3n-2)/ sqrt(pi*n^3).
D-finite with recurrence 10*n*(n-1)*(n-2)*a(n) -(n-1)*(n-2)*(149*n-396)*a(n-1) +2*(n-2)*(244*n^2-1618*n+2517)*a(n-2) +4
*(76*n^3-696*n^2+2165*n-2289)*a(n-3) +16*(2*n-9)*(56*n^2-336*n+451)*a(n-4) -256*(n-5)*(2*n-9)*(2*n-11)*a(n-5)=0. - R. J. Mathar, Jun 19 2021

a(0) removed from definition by Georg Fischer, Jun 19 2021

A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025

Offset: 0

Seiichi Manyama, Nov 03 2023

nonn

Cf. A176697, A367041.
Cf. A366266, A366676.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475, A364478.

a(n) = sum(k=0, n\2, binomial(2*(n-2*k)+1, k)*binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(2*(n-2*k)+1,k) * binomial(3*(n-2*k),n-2*k)/(2*(n-2*k)+1).

A218182 Number of truth tables of bracketed formulas (case 1).

Original entry on oeis.org

0, 0, 1, 6, 33, 194, 1198, 7676, 50581, 340682, 2335186, 16237284, 114255994, 812107412, 5822171548, 42052209400, 305714145869, 2235262899418, 16426616425002, 121265916776148, 898878250833358, 6687497426512700, 49920590244564484

Offset: 0

N. J. A. Sloane, Oct 23 2012

nonn

Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv preprint arXiv:1205.5595 [math.CO], 2012.

Cf. A186997, A218045.

my(x='x+O('x^30)); concat([0,0], Vec(((1-8*x)^(1/2)-3)*((2+2*(1-8*x)^(1/2)+8*x)^(1/2)-2)/8)) \\ Michel Marcus, Oct 21 2020

Yildiz gives a g.f.

G.f.: ((1-8*x)^(1/2)-3)*((2+2*(1-8*x)^(1/2)+8*x)^(1/2)-2)/8. - Mark van Hoeij, May 16 2013

A240586 Expansion of (((8-8 / sqrt(1-8x)) / (2sqrt(8x+2sqrt(1-8x)+2))+4 / sqrt(1-8x))((x(sqrt(8x+2sqrt(1-8x)+2)-sqrt(1-8x)-1))-4x^2)) / (sqrt(8x+2sqrt(1-8x)+2)-sqrt(1-8*x)-1)^2.

Original entry on oeis.org

1, 4, 22, 140, 950, 6692, 48284, 354216, 2630310, 19713188, 148817524, 1130011896, 8621650492, 66043991080, 507628779896, 3913088587472, 30240258982662, 234210742764964, 1817484391184900, 14128074297880536, 109992814064010196, 857525947713607096, 6693820044841440008

Offset: 1

Vladimir Kruchinin, Apr 08 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Rest[CoefficientList[Series[(((8-8 / Sqrt[1-8*x]) / (2*Sqrt[8*x+2*Sqrt[1-8*x]+2])+4 / Sqrt[1-8*x])*((x*(Sqrt[8*x+2*Sqrt[1-8*x]+2]-Sqrt[1-8*x]-1))-4*x^2)) / (Sqrt[8*x+2*Sqrt[1-8*x]+2]-Sqrt[1-8*x]-1)^2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 12 2014 *)

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

my(x='x+O('x^50)); Vec((((8-8 / sqrt(1-8*x)) / (2*sqrt(8*x+2*sqrt(1-8*x)+2))+4 / sqrt(1-8*x))*((x*(sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1))-4*x^2)) / (sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1)^2) \\ G. C. Greubel, Apr 05 2017

a(n) = Sum_{m=1..n} binomial(n-1, m-1)*Sum_{j=0..m} j*(-1)^(j-m)*binomial(m, j)*Sum_{k=1..n} binomial(k, n-k) * binomial(2*k+j-1, k+j-1) / (k+j).
A(x) = B'(x) * (x*B(x)-x^2) / B(x)^2, where B(x) = (-1-sqrt(1-8*x)+sqrt(2+2*sqrt(1-8*x)+8*x))/4, B(x)/x is g.f. of A186997.
a(n) ~ 8^(n-1) * (sqrt(3)-1) / sqrt(Pi*n). - Vaclav Kotesovec, Apr 12 2014

cp's OEIS Frontend

A192479 a(n) = 2^n*C(n-1) - A186997(n-1), where C(n) are the Catalan numbers (A000108).

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A364475 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^3.

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A218045 Number of truth tables of bracketed formulas (case 3).

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A364474 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x).

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A364478 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^8.

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A378290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n+2*r+k,r) * binomial(r,n-r)/(n+2*r+k) for k > 0.

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A192481 a(n) = Sum_{i=1..n-1} (2^i*C(i)-a(i)) * (2^(n-i)*C(n-i)-a(n-i)), a(1)=1, where C(i)=A000108(i-1) are Catalan numbers.

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A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

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A364475 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^3.

A364474 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x).

A364478 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^8.

A378290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n+2r+k,r) binomial(r,n-r)/(n+2*r+k) for k > 0.

A192481 a(n) = Sum_{i=1..n-1} (2^iC(i)-a(i)) (2^(n-i)*C(n-i)-a(n-i)), a(1)=1, where C(i)=A000108(i-1) are Catalan numbers.

A240586 Expansion of (((8-8 / sqrt(1-8x)) / (2sqrt(8x+2sqrt(1-8x)+2))+4 / sqrt(1-8x))((x(sqrt(8x+2sqrt(1-8x)+2)-sqrt(1-8x)-1))-4x^2)) / (sqrt(8x+2sqrt(1-8x)+2)-sqrt(1-8*x)-1)^2.