A024492 -id:A024492 - cp's OEIS frontend

A228332 Let h(m) denote the sequence whose n-th term is Sum_{k=0..n} (k+1)^m*T(n,k)^2, where T(n,k) is the Catalan triangle A039598. This is h(6).

1, 68, 1778, 43080, 958430, 20119736, 405350788, 7921691280, 151231519350, 2834134359000, 52320693313020, 953960351550960, 17212782834351468, 307826474156801840, 5462948893700675720, 96303960593503261984, 1687752152779483045542, 29424712141610821296408, 510621541414656188646220

Offset: 0

N. J. A. Sloane, Aug 26 2013

nonn

Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Remark 3 p. 1882. Omega6(n) = a(n-1).
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Cf. A000108, A039598, A024492, A000894, A228329, A000515, A228330, A228331, A228333.

Table[Sum[(k+1)^6*(Binomial[2n+1, n-k]*2*(k+1)/(n+k+2))^2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 08 2013 *)

Recurrence: n*(2*n+1)*(105*n^5 - 420*n^4 + 588*n^3 - 356*n^2 + 96*n - 10)*a(n) = 2*(4*n-7)*(4*n-5)*(105*n^5 + 105*n^4 - 42*n^3 - 62*n^2 - 7*n + 3)*a(n-1). - Vaclav Kotesovec, Dec 08 2013

a(n) = binomial(4*n,2*n) * (105*n^5 + 105*n^4 - 42*n^3 - 62*n^2 - 7*n + 3) / ((2*n+1)*(4*n-3)*(4*n-1)). - Vaclav Kotesovec, Dec 08 2013

A187359 Catalan trisection: A000108(3*n + 2)/2, n>=0.

Original entry on oeis.org

1, 21, 715, 29393, 1337220, 64822395, 3282060210, 171529806825, 9183676536076, 501121108325684, 27767032438524099, 1558142747453650631, 88366931393503350700, 5056959295818949067010, 291650059796498346544020, 16934386878595523443214745, 989130828878080326811887228, 58078935727891217125276922940, 3426228463922436748774829232156, 202972497563788492865321721683556

Offset: 0

Wolfdieter Lang, Mar 09 2011

See the comment under A187357 for the o.g.f.s of the general trisection of a sequence.

The sequence C(3*n+2) starts as 2, 42, 1430, 58786, 2674440, 129644790, 6564120420, 343059613650, ...

Cf. A000108, A024492, A048990, A187357 (C(3*n)), A187358 (C(3*n+1)).

Table[CatalanNumber[3*n+2]/2, {n, 0, 20}] (* Amiram Eldar, Mar 16 2022 *)

a(n) = C(3*n+2)/2, n>=0, with C(n) = A000108(n).
O.g.f.: (3 - sqrt(1 - 4*x^(1/3)) - sqrt(2)*sqrt(sqrt(1 + 4*x^(1/3) + 16*x^(2/3)) +
  (1 + 2*x^(1/3))))/(12*x).
From Ilya Gutkovskiy, Jan 21 2017: (Start)
E.g.f.: 3F3(5/6,7/6,3/2; 4/3,5/3,2; 64*x).
a(n) ~ 8^(2*n+1)/(3*sqrt(3*Pi)*n^(3/2)). (End)
Sum_{n>=0} a(n)/4^n = 1 - sqrt(3+2*sqrt(3))/3. - Amiram Eldar, Mar 16 2022
a(n) = (1/2)*Product_{1 <= i <= j <= 3*n+1} (3*i + j + 2)/(3*i + j - 1). - Peter Bala, Feb 22 2023

A230061 Primes of the form Catalan(n)+1.

Original entry on oeis.org

2, 3, 43, 58787, 4861946401453, 337485502510215975556783793455058624701, 4180080073556524734514695828170907458428751314321, 1000134600800354781929399250536541864362461089950801, 944973797977428207852605870454939596837230758234904051

Offset: 1

K. D. Bajpai, Oct 08 2013

nonn

The 25th term a(25) in the sequence has 693 digits.

a(26) has 1335 digits; a(27) has 1647 digits; a(28) has 1694 digits; a(29) has 2554 digits; a(30) has 4857 digits; a(31) has 4876 digits; a(32) has 9641 digits. - Charles R Greathouse IV, Oct 09 2013

			a(3)= 43: Catalan(5)= (2*5)!/(5!*(5+1)!)= 42. Catalan(5)+1= 43 which is prime.
a(4)= 58787: Catalan(11)= (2*11)!/(11!*(11+1)!)= 58786. Catalan(11)+1= 58787 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..25
Chris K. Caldwell and G. L. Honaker,Jr., Prime Curios! 4250

Cf. A000108, A024492, A141351.

Cf. A053429 (numbers n such that Catalan(n)+1 is prime).

KD:= proc() local a,b,c; a:= (2*n)!/(n!*(n + 1)!); b:=a+1;if isprime(b) then return(b): fi; end: seq(KD(),n=1..50);

Select[CatalanNumber[Range[100]]+1,PrimeQ] (* Harvey P. Dale, Aug 26 2021 *)

for(n=1,1e3,if(ispseudoprime(t=binomial(2*n,n)/(n+1)+1),print1(t", "))) \\ Charles R Greathouse IV, Oct 08 2013

A343386 Number of odd Motzkin n-paths, i.e., Motzkin n-paths with an odd number of up steps.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 20, 56, 168, 456, 1137, 2827, 7458, 20670, 57577, 157691, 427976, 1170552, 3248411, 9096497, 25505562, 71436182, 200338074, 564083786, 1595055520, 4522769520, 12842772295, 36514010301, 103995490758, 296794937626, 848620165860, 2430089817720

Offset: 0

Gennady Eremin, Sergey Kirgizov, Apr 13 2021

a(n) is the number of Motzkin n-paths with an odd number of U-steps (see A001006). For example, there are 9 Motzkin 4-paths, of which six have one U-step each, namely: 00UD, 0U0D, 0UD0, U00D, U0D0, and UD00. So a(4) = 6.

Number of Motzkin n-paths that, after removing the horizontal steps, are converted to Dyck (2m)-paths, where 2m <= n and m is odd (see A024492).

			G.f. = x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 56*x^7 + 168*x^8 + ...

Gennady Eremin, Table of n, a(n) for n = 0..800
Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021.

Cf. A000108, A001006, A107587, A024492, A100223.

a[n_] := ((n - 1) n HypergeometricPFQ[{1/2 - n/4, 3/4 - n/4, 1 - n/4, 5/4 - n/4}, {3/2, 3/2, 2}, 16])/2;
Table[a[n], {n, 0, 31}] (* Peter Luschny, Sep 24 2021 *)

M = [4, 9]; E = [1, 1, 1, 1, 3];
A343386 = [0, 0, 1, 3, 6]
for n in range(5, 801):
    M.append(((2*n+1)*M[1]+(3*n-3)*M[0])//(n+2))
    E.append(((5*n**2+n-3)*E[4] - (10*n**2-16*n+3)*E[3]
      + (10*n**2-34*n+27)*E[2] + (11*n-5)*(n-3)*E[1]
      - 15*(n-3)*(n-4)*E[0]) // (n*n+2*n))
    A343386.append(M[-1] - E[-1])
    M.pop(0); E.pop(0)

a(n) = Sum_{k=0..n} binomial(n, 4*k+2) * A000108(2*k+1).
a(n) = A001006(n) - A107587(n).
G.f.: A(x) = (2 - 2*x - sqrt(1-2*x-3*x^2) - sqrt(1-2*x+5*x^2))/(4*x^2).
G.f. A(x) satisfies A(x) = x*A(x) + x^2*A(x)^2 + x^2*B(x)^2 where B(x) is the g.f. of A107587.
a(n) = A107587(n) - A100223(n+2). - R. J. Mathar, Apr 16 2021
D-finite with recurrence: n*(n+2)*a(n) + (-5*n^2-n+3)*a(n-1) + (10*n^2-16*n+3)*a(n-2) + (-10*n^2+34*n-27)*a(n-3) - (11*n-5)*(n-3)*a(n-4)  + 15*(n-3)*(n-4)*a(n-5) = 0, n >= 5. - R. J. Mathar, Apr 17 2021
D-finite with recurrence: n*(n-2)*(n+2)*a(n) - (2*n-1)*(2*n^2-2*n-3)*a(n-1) + 3*(n-1)*(2*n^2-4*n+1)*a(n-2) - 2*(n-1)*(n-2)*(2*n-3)*a(n-3) - 15*(n-1)*(n-2)*(n-3)*a(n-4) = 0, n >= 4. - R. J. Mathar, Apr 17 2021
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * A000108(k) * (k mod 2). - Gennady Eremin, May 03 2021 [after Paul Barry (A107587)]
a(n) = ((n-1)*n*hypergeom([1/2-n/4, 3/4-n/4, 1-n/4, 5/4-n/4], [3/2, 3/2, 2], 16))/2. - Peter Luschny, Sep 24 2021
a(n) ~ 3^(n + 3/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 27 2024

A024491 a(n) = (1/(4n-1))*C(4n,2n).

Original entry on oeis.org

-1, 2, 10, 84, 858, 9724, 117572, 1485800, 19389690, 259289580, 3534526380, 48932534040, 686119227300, 9723892802904, 139067101832008, 2004484433302736, 29089272078453818, 424672260824486220, 6232570989814602524, 91901608649243484728, 1360850743459951600780

Offset: 0

Clark Kimberling

			sqrt(1/2*(1+sqrt(1-x))) = 1 - 1/8*x - 5/128*x^2 - 21/1024*x^3 - ...

Cf. A000108, A001448, A024492, A245112.

[(1/(4*n-1))*Binomial(4*n,2*n) : n in [0..20]]; // Wesley Ivan Hurt, Jan 06 2024

Table[1/(4n-1) Binomial[4n,2n],{n,0,20}] (* or *) With[{c=4Sqrt[x]}, CoefficientList[ Series[(-Sqrt[1-c]-Sqrt[1+c])/2,{x,0,30}],x]] (* Harvey P. Dale, Mar 10 2013 *)

G.f.: A(x) = -sqrt((1/2)*(1+sqrt(1-16*x))).
With interpolated zeros, this has g.f. -(sqrt(1-4x)+sqrt(1+4x))/2. - Paul Barry, Dec 23 2006
D-finite with recurrence n*(2*n-1)*a(n) - 2*(4*n-3)*(4*n-5)*a(n-1) = 0. - R. J. Mathar, Nov 13 2012
a(n) = A001448(n)/(4*n-1). - R. J. Mathar, Apr 27 2020
From Peter Bala, Apr 02 2023: (Start)
O.g.f. A(x) = - sqrt(1 - 4*x*C(4*x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108.
The series reversion of -x*A(x) is equal to x * the o.g.f. of A245112. (End)
a(n) ~ 2^(4*n-5/2) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 04 2025

More terms from Harvey P. Dale, Mar 10 2013

A099976 Bisection of A000984.

Original entry on oeis.org

2, 20, 252, 3432, 48620, 705432, 10400600, 155117520, 2333606220, 35345263800, 538257874440, 8233430727600, 126410606437752, 1946939425648112, 30067266499541040, 465428353255261088, 7219428434016265740

Offset: 0

N. J. A. Sloane, Nov 19 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000108, A000265, A000984, A001448, A002458, A063079.

[Binomial(4*n+2, 2*n+1): n in [0..20]]; // Vincenzo Librandi, May 22 2011

seq(binomial(4*n+2,2*n+1),n=0..20); # Emeric Deutsch, Dec 20 2004

Array[Binomial[4*# + 2, 2*# + 1] &, 20, 0] (* Paolo Xausa, Jul 11 2024 *)

a(n) = binomial(4n+2, 2n+1). - Emeric Deutsch, Dec 20 2004
G.f.: 2*sqrt(2)/sqrt(1-16*x)/sqrt(1+sqrt(1-16*x)) = 2 + 60*x/(G(0)-30*x) where G(k)= 2*x*(4*k+3)*(4*k+5) + (2*k+3)*(k+1)- 2*x*(k+1)*(2*k+3)*(4*k+7)*(4*k+9)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 14 2012
G.f. A(x) satisfies A(x^2) = F'(x)/F(x), where F(x) = C(x)/C(-x) and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, May 15 2023
From R. J. Mathar, Jul 11 2024: (Start)
D-finite with recurrence n*(2*n+1)*a(n) -2*(4*n-1)*(4*n+1)*a(n-1)=0.
a(n) = 2*A002458(n).
G.f.: 2* 2F1(3/4,5/4; 3/2 ; 16*x).
Conjecture: A000265(a(n)) = A063079(n+1), odd part of a(n). (End)
a(n) / (2*n+2) = A024492(n). - R. J. Mathar, Jul 12 2024

More terms from Emeric Deutsch, Dec 20 2004

A382394 a(n) = Sum_{k=0..n} A128899(n,k)^3.

Original entry on oeis.org

1, 1, 9, 190, 5705, 204876, 8209278, 354331692, 16140234825, 765868074400, 37525317999884, 1886768082651816, 96906387191038334, 5066711735118128200, 268954195756648761900, 14464077426547576156440, 786729115199980286001225, 43219452658242723841261800

Offset: 0

Seiichi Manyama, Mar 24 2025

nonn

Let b_k(n) = Sum_{j=0..n} A128899(n,j)^k. b_1(n) = binomial(2*n-1,n) = A088218(n) and b_2(n) = A024492(n-1) for n > 0.

Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

Cf. A024492, A088218.

Cf. A001700, A003161, A128899, A183069.

a128899(n, k) = binomial(2*n-2, n-k)-binomial(2*n-2, n-k-2);
a(n) = sum(k=0, n, a128899(n, k)^3);

a(n) = binomial(2*n,n)/2 * A183069(n) for n > 0.

a(n) = A003161(2*n-1) for n > 0.

A133603 The matrix-vector product A133566 * A000108.

Original entry on oeis.org

1, 1, 3, 5, 19, 42, 174, 429, 1859, 4862, 21658, 58786, 266798, 742900, 3417340, 9694845, 45052515, 129644790, 607283490, 1767263190, 8331383610, 24466267020, 115948830660, 343059613650, 1632963760974, 4861946401452

Offset: 0

Gary W. Adamson, Sep 18 2007

A133602 is a companion sequence.

			a(5) = C(5) = 42.
a(6) = 174 = C(6) + C(5) = 132 + 42.

Cf. A133566, A000108, A133602, A024492 (bisection).

A133603 := proc(n)
    add(A133566(n+1,k+1)*A000108(k),k=0..n) ;
end proc: # R. J. Mathar, Jun 20 2015

A133566 * A000108 where A133566 = an infinite lower triangular matrix and A000108 = the Catalan sequence. For odd n, a(n) = C(n). For even n, a(n) = C(n) + C(n-1) = A005807(n-1).
Conjecture: n*(n-2)*(3*n-1)*(n+1)*a(n) -8*n*(2*n-3)*a(n-1) -4*(n-1)*(3*n+2)*(
2*n-3)*(2*n-5)*a(n-2)=0. - R. J. Mathar, Jun 20 2015

A234040 a(n) = binomial(2(n+1),n) gcd(n,2)/(2*(n+1)).

Original entry on oeis.org

1, 1, 5, 7, 42, 66, 429, 715, 4862, 8398, 58786, 104006, 742900, 1337220, 9694845, 17678835, 129644790, 238819350, 1767263190, 3282060210, 24466267020, 45741281820, 343059613650, 644952073662, 4861946401452, 9183676536076, 69533550916004

Offset: 0

Wolfdieter Lang, Feb 23 2014

This gives the next-to-central entries of the even-indexed rows of the triangle A107711.
For the central entries (of the even-numbered rows) see A001700.
This sequence is composed of the bisection sequences A024492 (even part) and A065097 (odd part).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A024492, A065097, A107711.

[Binomial(2*(n+1),n)*Gcd(n,2)/(2*(n+1)): n in [0..30]]; // Vincenzo Librandi, Feb 25 2014

Table[Binomial[2 (n + 1), n] GCD[n, 2]/(2 (n + 1)), {n, 0, 40}] (* Vincenzo Librandi, Feb 25 2014 *)

a(n) = binomial(2*(n+1),n)*gcd(n,2)/(2*(n+1)) for n >= 0.
a(n) = A107711(2*(n+1), n) for n >= 0.
G.f.: (3*c(x)- c(-x)-2)/(4*x) =(4*(1-x) - 3*sqrt(1-4*x) - sqrt(1+4*x))/(8*x^2), with c(x) the o.g.f. of the Catalan numbers A000108. See the bisection comment above.

a(26) from Vincenzo Librandi, Feb 25 2014

A276484 Decimal expansion of Sum_{k>=0} (2k+2)/binomial(4k+2, 2*k+1).

Original entry on oeis.org

1, 2, 2, 6, 3, 6, 4, 6, 7, 1, 2, 1, 6, 7, 4, 2, 7, 1, 3, 9, 0, 9, 9, 2, 5, 8, 1, 0, 9, 3, 9, 7, 3, 5, 4, 6, 4, 8, 3, 1, 6, 8, 9, 4, 6, 3, 3, 8, 5, 8, 3, 4, 0, 8, 9, 4, 9, 0, 5, 4, 4, 7, 8, 3, 9, 3, 3, 3, 5, 3, 1, 7, 6, 4, 0, 5, 4, 1, 6, 9, 7, 8, 2, 1, 2, 1, 1, 8, 7, 7, 2, 0, 1, 8, 9, 0, 1, 7, 1, 5, 7, 1, 0, 0, 1, 3, 2, 0, 0, 1, 5, 2, 6, 5, 9, 6, 8, 6, 9, 8

Offset: 1

Ilya Gutkovskiy, Sep 05 2016

			1.22636467121674271390992581093973546...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Catalan Number

Cf. A000108, A001622, A024492, A121839, A268813.

SetDefaultRealField(RealField(100)); R:=RealField(); 2*Pi(R)/(9*Sqrt(3)) + 6*(2*Sqrt(5)*Log((1+Sqrt(5))/2) + 15)/125; // G. C. Greubel, Nov 04 2018

RealDigits[2 (Pi/(9 Sqrt[3])) + 6 ((2 Sqrt[5] Log[GoldenRatio] + 15)/125), 10, 120][[1]]
RealDigits[HypergeometricPFQ[{1, 3/2, 2}, {3/4, 5/4}, 1/16], 10, 120][[1]]

suminf(k=0,(2*k+2)/binomial(4*k+2,2*k+1)) \\ Indranil Ghosh, Mar 04 2017

default(realprecision, 100); 2*Pi/(9*sqrt(3)) + 6*(2*sqrt(5)*log((1+sqrt(5))/2) + 15)/125 \\ G. C. Greubel, Nov 04 2018

Equals 2*Pi/(9*sqrt(3)) + 6*(2*sqrt(5)*log(phi) + 15)/125, where phi is the golden ratio (A001622).
Equals Sum_{k>=0} 1/Catalan number(2k+1).
Equals Sum_{k>=0} 1/A000108(2k+1).
Equals Sum_{k>=0} 1/A024492(k).

cp's OEIS Frontend

A228332 Let h(m) denote the sequence whose n-th term is Sum_{k=0..n} (k+1)^m*T(n,k)^2, where T(n,k) is the Catalan triangle A039598. This is h(6).

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A187359 Catalan trisection: A000108(3*n + 2)/2, n>=0.

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A230061 Primes of the form Catalan(n)+1.

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A343386 Number of odd Motzkin n-paths, i.e., Motzkin n-paths with an odd number of up steps.

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A024491 a(n) = (1/(4n-1))*C(4n,2n).

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Magma

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A099976 Bisection of A000984.

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A382394 a(n) = Sum_{k=0..n} A128899(n,k)^3.

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A133603 The matrix-vector product A133566 * A000108.

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A234040 a(n) = binomial(2(n+1),n) gcd(n,2)/(2*(n+1)).

A276484 Decimal expansion of Sum_{k>=0} (2k+2)/binomial(4k+2, 2*k+1).