A000984 -id:A000984 - cp's OEIS frontend

A106187 Sequence array for central binomial numbers A000984.

1, 2, 1, 6, 2, 1, 20, 6, 2, 1, 70, 20, 6, 2, 1, 252, 70, 20, 6, 2, 1, 924, 252, 70, 20, 6, 2, 1, 3432, 924, 252, 70, 20, 6, 2, 1, 12870, 3432, 924, 252, 70, 20, 6, 2, 1, 48620, 12870, 3432, 924, 252, 70, 20, 6, 2, 1, 184756, 48620, 12870, 3432, 924, 252, 70, 20, 6, 2, 1

Offset: 0

Paul Barry, Apr 24 2005

			Triangle begins:
    1;
    2,  1;
    6,  2, 1;
   20,  6, 2, 1;
   70, 20, 6, 2, 1;
  252, 70, 20, 6, 2, 1;
  ...
The matrix inverse starts:
   1;
  -2,1;
  -2,-2,1;
  -4,-2,-2,1;
  -10,-4,-2,-2,1;
  -28,-10,-4,-2,-2,1;
  -84,-28,-10,-4,-2,-2,1;
  -264,-84,-28,-10,-4,-2,-2,1;
apparently related to A002420. - _R. J. Mathar_, Apr 08 2013

Row sums are A006134.
Antidiagonal sums are A106188.
Cf. A000984.

A106187 := proc(n,k)
    binomial(2*(n-k),n-k) ;
end proc: # R. J. Mathar, Apr 08 2013

T[n_, k_] := (((2*n - 2*k)!)/((n - k)!)^2); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Detlef Meya, Aug 11 2024 *)

T(n, k) = binomial(2*(n-k), n-k).

Riordan array (1/sqrt(1-4x), x).

A134762 a(n) = 3*A000984(n) - 2.

Original entry on oeis.org

1, 4, 16, 58, 208, 754, 2770, 10294, 38608, 145858, 554266, 2116294, 8112466, 31201798, 120349798, 465352558, 1803241168, 7000818658, 27225405898, 106035791398, 413539586458, 1614773623318, 6312296891158, 24700292182798, 96742811049298, 379231819313254

Offset: 0

Gary W. Adamson, Nov 09 2007

nonn

Second inverse binomial transform of the sequence = A134763, (same as a(n) but with interpolated two's).

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

Cf. A000108, A000984, A134758, A134759, A134760, A134761, A134763, A134770, A134771.

[3*(n+1)*Catalan(n)-2: n in [0..40]]; // G. C. Greubel, May 28 2024

Table[3*Binomial[2*n,n]-2, {n,0,40}] (* G. C. Greubel, May 28 2024 *)

a(n) = 3*binomial(2*n, n) - 2; \\ Michel Marcus, Nov 22 2013

[3*binomial(2*n,n) -2 for n in range(41)] # G. C. Greubel, May 28 2024

G.f.: 3/sqrt(1-4*x) - 2/(1-x). - Sergei N. Gladkovskii, Nov 21 2013
From G. C. Greubel, May 28 2024: (Start)
a(n) = 3*(n+1)*A000108(n) - 2.
a(n) = (2*(2*n-1)*a(n-1) + 2*(3*n-2))/n.
E.g.f.: 3*exp(2*x)*BesselI(0, 2*x) - 2*exp(x). (End)

More terms from Michel Marcus, Nov 22 2013

A163841 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984).

Original entry on oeis.org

1, 3, 2, 11, 8, 6, 45, 34, 26, 20, 195, 150, 116, 90, 70, 873, 678, 528, 412, 322, 252, 3989, 3116, 2438, 1910, 1498, 1176, 924, 18483, 14494, 11378, 8940, 7030, 5532, 4356, 3432, 86515, 68032, 53538

Offset: 0

Peter Luschny, Aug 06 2009

For n >= 0, k >= 0 let T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040). Triangle read by rows.

			Triangle begins
     1;
     3,    2;
    11,    8,    6;
    45,   34,   26,   20;
   195,  150,  116,   90,   70;
   873,  678,  528,  412,  322,  252;
  3989, 3116, 2438, 1910, 1498, 1176,  924;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Row sums are A163844. Cf. A056040, A163650, A163841, A163842, A163840, A026375, A002426, A000984.

Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
a := n -> SumTria(k->swing(2*k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

A181418 a(n) = A000984(n)*A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively.

Original entry on oeis.org

1, 4, 60, 1120, 24220, 567504, 14030016, 360222720, 9513014940, 256758913840, 7051260776560, 196403499277440, 5535202897806400, 157551884911456000, 4522682234563776000, 130783762623673221120, 3806221127760278029980

Offset: 0

Paul D. Hanna, Jan 28 2011

This sequence is s_6 in Cooper's paper. - Jason Kimberley, Nov 25 2012

Diagonal of the rational function R(x,y,z,w)=1/(1-(w*x*y+w*z+x*y+x*z+y+z)). - Gheorghe Coserea, Jul 13 2016

			E.g.f.: A(x) = 1 + 4*x/2! + 60*x^2/(2!*4!) + 1120*x^3/(3!*6!) + 24220*x^4/(4!*8!) + 567504*x^5/(5!*10!) +....
where A(x)^(1/2) = 1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 +x^5/5!^3 +...

Jason Kimberley, Table of n, a(n) for n = 0..226
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.

Cf. A000984, A000172, A199813.

Related to diagonal of rational functions: A268545-A268555.

Table[Binomial[2n,n]*Sum[Binomial[n,k]^3,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 06 2014 *)

{a(n)=binomial(2*n,n)*sum(k=0,n,binomial(n,k)^3)}

{a(n)=(2*n)!*n!*polcoeff(sum(m=0, n, x^m/m!^3+x*O(x^n))^2, n)}

a(n) = C(2n,n) * Sum_{k=0..n} C(n,k)^3.
E.g.f.: Sum_{n>=0} a(n)*x^n/(n!*(2*n)!) = ( Sum_{n>=0} x^n/n!^3 )^2.
From Jason Kimberley, Nov 26 2012: (Start)
1/Pi
  = (2/25)*Sum_{n>=0} a(n)*(9n+2)/50^n. [Cooper, equation (5)]
  = (2/25)*Sum_{n>=0} a(n)*A017185(n)/A165800(n). (End)
G.f.: 4*hypergeom([1/6, 1/3],[1],(27/2)*(1+(1-32*x)^(1/2))*(1-(1-32*x)^(1/2))^2/(3+(1-32*x)^(1/2))^3)^2/(3+(1-32*x)^(1/2)). - Mark van Hoeij, May 07 2013
Recurrence: n^3*a(n) = 2*(2*n-1)*(7*n^2 - 7*n + 2)*a(n-1) + 32*(n-1)*(2*n-3)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Mar 06 2014
a(n) ~ 2^(5*n+1) / (sqrt(3) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Mar 06 2014
0 = (-x^2+28*x^3+128*x^4)*y''' + (-3*x+126*x^2+768*x^3)*y'' + (-1+92*x+864*x^2)*y' + (4+96*x)*y, where y is g.f. - Gheorghe Coserea, Jul 13 2016

A203576 Exponential (or binomial) half-convolution of A000984 (central binomial) with itself.

Original entry on oeis.org

1, 2, 14, 56, 446, 2152, 18248, 97120, 848254, 4796552, 42454664, 250140640, 2226532712, 13516860320, 120553738144, 748819997056, 6679690686334, 42254745008840, 376638926040392, 2418457241945056, 21530200591563496, 139992790135717792, 1244418656720926624, 8178446389043428736

Offset: 0

Wolfdieter Lang, Jan 13 2012

In general the exponential (also known as binomial) half-convolution of a sequence {b(n), n>=0} with itself is defined by
bhat(n) := Sum_{k=0..floor(n/2)} binomial(n,k)*b(k)*b(n-k), n>=0.
The e.g.f. of the sequence {bhat(n)} is Bhat(x) = ((B(x))^2 + B2(x^2))/2, with the e.g.f. B(x) of {b(n), n>=0} and the e.g.f. B2(x) := Sum_((b(n)^2/n!)*x^n/n!, n>=0) of the scaled squares. The proof runs along the same line as the one given for the ordinary half-convolution in a comment on A201204. In fact, bhat(n)/n! is the ordinary half-convolution of the sequence {b(n)/n!, n>=0} with itself.
Here b(n) = A000984(n) = binomial(2*n,n), n>=0, B(x) = exp(2*x)*BesselI(0,2*x) (see the Abramowitz-Stegun reference and link under A008277 for BesselI, p. 375, eq. 9.6.10) and B2(x) = hypergeometric([1/2,1/2],[1,1,1],16*x).

			With cbi = {1, 2, 6, 20, 70, 252, ...}
a(4) = 1*70 + 4*2*20 + 6*6^2 = 446,
a(5) = 1*252 + 5*2*70 + 10*6*20 = 2152.

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

Cf. A000984, A081085 (exponential convolution).

cbi[n_] := Binomial[2*n, n]; a[n_] := Sum[Binomial[n, k]*cbi[k]*cbi[n - k], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 09 2013 *)

A203576(n)=sum(k=0,n\2,binomial(n,k)*A000984(k)*A000984(n-k))  \\ M. F. Hasler, Jan 13 2012

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*cbi(k)*cbi(n-k) n>=0, with cbi(n)=A000984(n).
E.g.f.: (exp(4*x)*BesselI(0, 2*x)^2 + hypergeom([1/2,1/2], [1,1,1],(4*x)^2))/2. See comment above.
Recurrence: (n-1)^2 * n^3 * (3*n^5 - 40*n^4 + 200*n^3 - 476*n^2 + 544*n - 241)*a(n) = 4*(n-1)^3 * (9*n^7 - 126*n^6 + 699*n^5 - 1997*n^4 + 3165*n^3 - 2770*n^2 + 1239*n - 228)*a(n-1) + 32*(3*n^10 - 55*n^9 + 420*n^8 - 1786*n^7 + 4731*n^6 - 8232*n^5 + 9630*n^4 - 7580*n^3 + 3900*n^2 - 1194*n + 162)*a(n-2) - 256*(n-2)^3 * (9*n^7 - 126*n^6 + 699*n^5 - 1997*n^4 + 3165*n^3 - 2770*n^2 + 1239*n - 228)*a(n-3) + 2048*(n-3)^3 * (n-2)^2 * (3*n^5 - 25*n^4 + 70*n^3 - 86*n^2 + 47*n - 10)*a(n-4). - Vaclav Kotesovec, Feb 25 2014
a(n) ~ 8^n / (Pi*n) * (1 + (1+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, Feb 25 2014

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

A135757 Central binomial coefficients at triangular positions: a(n) = A000984(n(n+1)/2).

Original entry on oeis.org

1, 2, 20, 924, 184756, 155117520, 538257874440, 7648690600760440, 442512540276836779204, 103827421287553411369671120, 98527218530093856775578873054432, 377389666165540953244592352291892721700, 5825874245311064218315521996517139009907512400

Offset: 0

Paul D. Hanna, Dec 02 2007

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A000984, A135758.

[Binomial(n*(n+1), n*(n+1) div 2): n in [0..15]]; // Vincenzo Librandi, Nov 08 2016

seq(binomial(n*(n+1),n*(n+1)/2),n=0..20); # Robert Israel, Nov 08 2016

Table[Binomial[n*(n + 1), n*(n + 1)/2], {n,0,10}] (* G. C. Greubel, Nov 07 2016 *)

PARI
```
a(n)=binomial(n*(n+1),n*(n+1)/2)
```

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

a(n) ~ 2^(n^2+n) sqrt(2/Pi) (1/n - 1/(2n^2) + 1/(8n^3) + ...). - Robert Israel, Nov 08 2016

A192655 Floor-Sqrt transform of central binomial coefficients (A000984).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 30, 58, 113, 220, 429, 839, 1644, 3224, 6333, 12454, 24516, 48307, 95263, 188003, 371276, 733660, 1450551, 2869395, 5678697, 11243247, 22269228, 44124136, 87456792, 173399153, 343896178, 682223096, 1353744488, 2686899408, 5334139244, 10591802387

Offset: 0

Emanuele Munarini, Jul 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000196, A000984.

[Floor(Sqrt(Binomial(2*n,n))): n in [0..40]]; // Vincenzo Librandi, Apr 01 2012

Table[Floor[Sqrt[Binomial[2n,n]]],{n,0,100}]

makelist(floor(sqrt(binomial(2*n,n))),n,0,24);

a(n) = sqrtint(binomial(2*n,n)) \\ Jason Yuen, Oct 21 2024

a(n) = floor(sqrt(binomial(2*n,n))).

a(n) = A000196(A000984(n)). - Jason Yuen, Oct 21 2024

A205946 Triangle read by rows related to A000984, central binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 6, 1, 1, 33, 27, 8, 1, 1, 88, 108, 44, 10, 1, 1, 232, 405, 208, 65, 12, 1, 1, 609, 1459, 908, 350, 90, 14, 1

Offset: 0

Gary W. Adamson, Feb 01 2012

			First few rows of the triangle =
1;
1, 1;
1, 4, 1;
1, 12, 6, 1;
1, 33, 27, 8, 1;
1, 88, 108, 44, 10, 1;
1, 232, 405, 208, 65, 12, 1;
1, 609, 1459, 908, 350, 90, 14, 1;
...
Row 2 = (1, 4, 1) = row 4 of triangle A191314.
Row 3 = (1, 12, 6, 1) = finite differences of column 6 of the array shown in A205573: (1, 13, 19, 20)

Cf. A000984 (row sums), A001405, A191314, A205573, A205945 (companion).

T(n,k) = A191314(2*n,k).

Take finite differences of even numbered columns of the A205573 array from the top -> down.

A258290 Arithmetic derivative of central binomial coefficients, cf. A000984.

Original entry on oeis.org

0, 1, 5, 24, 59, 456, 1448, 6868, 19749, 69364, 236356, 1526956, 3717440, 22858340, 122553540, 474051984, 954720543, 5726109024, 19329586520, 92051285020, 319059863484, 1271796704788, 4829219746964, 29791326914640, 74372011398840, 340296661452300

Offset: 0

Reinhard Zumkeller, May 26 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000984, A003415, A258197.

Haskell
```
a258290 = a003415 . a000984
```

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; a[n_] := ad[Binomial[2*n, n]]; Array[a, 26, 0] (* Amiram Eldar, Apr 13 2025 *)

a(n) = A003415(A000984(n)).

Central terms in triangle A258197: a(n) = A258197(2*n,n).

cp's OEIS Frontend

A106187 Sequence array for central binomial numbers A000984.

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A134762 a(n) = 3*A000984(n) - 2.

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A163841 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984).

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A181418 a(n) = A000984(n)*A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively.

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A203576 Exponential (or binomial) half-convolution of A000984 (central binomial) with itself.

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Mathematica

PARI

Formula

A099976 Bisection of A000984.

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Magma

Maple

Mathematica

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A135757 Central binomial coefficients at triangular positions: a(n) = A000984(n(n+1)/2).

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