A000984 -id:A000984 - cp's OEIS frontend

A157004 Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.

1, 2, 6, 18, 58, 192, 650, 2232, 7746, 27096, 95376, 337404, 1198546, 4272308, 15273888, 54744268, 196646922, 707747988, 2551624304, 9213416524, 33313656888, 120604436624, 437112790668, 1585877246424, 5759085911154

Offset: 0

Paul Barry, Feb 20 2009

Hankel transform is A157005. Image of A000984 under Riordan array (1,x(1-x^2)).

Diagonal of rational function 1/(1 - x - y + x^3*y^2). - Seiichi Manyama, Mar 23 2023

			G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 192*x^5 + 650*x^6 + 2232*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)

Cf. A157003, A360219, A360266.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-4*x+4*x^3) )); // G. C. Greubel, Feb 26 2019

CoefficientList[Series[1/Sqrt[1-4*x*(1-x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

{a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m)*(1-x)^m / (1-2*x+x*O(x^n))^(2*m+1)), n)} \\ Paul D. Hanna, Sep 21 2013

my(x='x+O('x^30)); Vec(1/sqrt(1-4*x+4*x^3)) \\ G. C. Greubel, Feb 26 2019

(1/sqrt(1-4*x+4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

G.f.: 1/sqrt(1 - 4*x*(1 - x^2)).
a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2)) *A000984(k)/2.
G.f.: Sum_{n>=0} (2*n)!/n!^2 * x^(2*n) * (1-x)^n / (1-2*x)^(2*n+1). - Paul D. Hanna, Sep 21 2013
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 2*(2*n-3)*a(n-3). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ (1/r)^n / (sqrt(Pi*n) * sqrt(3-8*r)), where r = 0.2695944364054... is the root of the equation 4*r*(1-r^2)=1. - Vaclav Kotesovec, Feb 13 2014
0 = a(n)*(16*a(n+1) - 32*a(n+3) + 10*a(n+4)) + a(n+1)*(-2*a(n+3)) + a(n+2)*(16*a(n+3) - 6*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Sep 03 2016

A038665 Convolution of A007054 (super ballot numbers) with A000984 (central binomial coefficients).

Original entry on oeis.org

3, 8, 25, 84, 294, 1056, 3861, 14300, 53482, 201552, 764218, 2912168, 11143500, 42791040, 164812365, 636438060, 2463251010, 9552774000, 37112526990, 144410649240, 562724141460, 2195581527360, 8576490341250, 33537507830424

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A000108, A000777, A000984, A001622, A007054.

[(n+3)*Catalan(n+1): n in [0..30]]; // Vincenzo Librandi, Sep 11 2016

seq((n+3)*binomial(2*n+2, n+1)/(n+2), n=0..24); # Zerinvary Lajos, Dec 08 2008

Table[(n + 3) (CatalanNumber[n + 1]), {n, 0, 30}] (* Vincenzo Librandi, Sep 11 2016 *)

a(n) = (n+3)*C(n+1) with C(n) the Catalan numbers A000108.
G.f.: c(x)*(4 - c(x))/sqrt(1 - 4*x) with c(x) the g.f. for the Catalan numbers.
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=0} 1/a(n) = 41/6 - 64*Pi/(9*sqrt(3)) + 2*Pi^2/3.
Sum_{n>=0} (-1)^n/a(n) = 57/10 - 256*log(phi)/(5*sqrt(5)) + 24*log(phi)^2, where phi is the golden ratio (A001622). (End)

A053214 Central binomial coefficients (A000984) read mod 2n, with a(0)=1.

Original entry on oeis.org

1, 0, 2, 2, 6, 2, 0, 2, 6, 2, 16, 2, 4, 2, 20, 0, 6, 2, 24, 2, 20, 6, 28, 2, 12, 2, 32, 20, 0, 2, 4, 2, 6, 42, 40, 42, 52, 2, 44, 20, 20, 2, 0, 2, 48, 0, 52, 2, 60, 2, 56, 54, 96, 2, 60, 32, 88, 96, 64, 2, 96, 2, 68, 12, 70, 70, 0, 2, 36, 66, 40, 2, 36, 2, 80, 120, 32, 0, 144, 2, 20, 20, 88

Offset: 0

Asher Auel, Dec 16 1999

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A059288, A059289.

a053214 0 = 1
a053214 n = a053200 (2 * n) n  -- Reinhard Zumkeller, Jan 24 2014

Join[{1}, Table[Mod[Binomial[2*n, n], 2*n], {n, 1, 100}]] (* G. C. Greubel, Sep 04 2018 *)

concat([1], vector(100, n, lift(Mod(binomial(2*n,n), 2*n)))) \\ G. C. Greubel, Sep 04 2018

a(n) = binomial(2*n, n) mod 2*n, with a(0)=1.

a(n) = A053200(2*n,n) for n > 0. - Reinhard Zumkeller, Jan 01 2013

More terms from James Sellers, Dec 18 1999

A134758 a(n) = A000984(n) + n.

Original entry on oeis.org

1, 3, 8, 23, 74, 257, 930, 3439, 12878, 48629, 184766, 705443, 2704168, 10400613, 40116614, 155117535, 601080406, 2333606237, 9075135318, 35345263819, 137846528840, 538257874461, 2104098963742, 8233430727623, 32247603683124, 126410606437777, 495918532948130

Offset: 0

Gary W. Adamson, Nov 09 2007

nonn

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

Cf. A000984, A134759, A134760, A134761, A134762, A134763, A134770, A134771.

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

Table[Binomial[2n,n]+n,{n,0,40}] (* Harvey P. Dale, Dec 10 2011 *)

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

G.f.: ((1-x)^2 + x*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x)). -  Harvey P. Dale, Dec 10 2011
From G. C. Greubel, May 28 2024: (Start)
E.g.f.: x*exp(x) + exp(2*x)*BesselI(0, 2*x).
a(n) = (2*(2*n-1)*a(n-1) - (3*n^2 - 6*n + 2))/n. (End)

More terms from Harvey P. Dale, Dec 10 2011

A134770 a(n) = 4*A000984(n) - 3.

Original entry on oeis.org

1, 5, 21, 77, 277, 1005, 3693, 13725, 51477, 194477, 739021, 2821725, 10816621, 41602397, 160466397, 620470077, 2404321557, 9334424877, 36300541197, 141381055197, 551386115277, 2153031497757, 8416395854877, 32933722910397, 128990414732397, 505642425751005, 1983674131792413

Offset: 0

Gary W. Adamson, Nov 10 2007

nonn

The second inverse binomial transform of this sequence is A134771, the sequence interleaved with threes: (1, 3, 5, 3, 21, 3, 77, 3, ...).

			a(2) = 21 = 4*A000984(2) - 3 = 4*6 - 3.

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

Cf. A000108, A000984, A134771.

[4*(n+1)*Catalan(n)-3: n in [0..40]]; // G. C. Greubel, Oct 13 2023

Table[4 Binomial[2n,n]-3,{n,0,30}] (* Harvey P. Dale, Dec 01 2022 *)

a(n)=4*binomial(2*n, n) - 3; \\ Michel Marcus, Jul 02 2020

[4*binomial(2*n,n)-3 for n in range(41)] # G. C. Greubel, Oct 13 2023

From G. C. Greubel, Oct 13 2023: (Start)
a(n) = 4*(n+1)*A000108(n) - 3.
G.f.: 4/sqrt(1-4*x) - 3/(1-x).
Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = 4*BesselI(0, 2*x) - cosh(x). (End)

a(10) corrected and offsets aligned by Georg Fischer, Jul 01 2020

More terms from Michel Marcus, Jul 02 2020

A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870

Offset: 0

Peter Luschny, Aug 05 2009

Triangle read by rows. For n >= 0, k >= 0 let T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040).

This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019

			Triangle begins
    1;
    1,   2;
    3,   4,   6;
    7,  10,  14,  20;
   19,  26,  36,  50,  70;
   51,  70,  96, 132, 182, 252;
  141, 192, 262, 358, 490, 672, 924;
From _M. F. Hasler_, Nov 15 2019: (Start)
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:
     1   1    3    7    19    51 ...
     2   4   10   26    70   192 ...
     6  14   36   96   262   726 ...
    20  50  132  358   988  2760 ...
    70 182  490 1346  3748 10540 ...
   252 672 1836 5094 14288 40404 ...
  (...)
Read by falling antidiagonals this yields the same sequence. (End)

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.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Row sums are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*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 *)

A115255 "Correlation triangle" of central binomial coefficients A000984.

Original entry on oeis.org

1, 2, 2, 6, 5, 6, 20, 14, 14, 20, 70, 46, 41, 46, 70, 252, 160, 134, 134, 160, 252, 924, 574, 466, 441, 466, 574, 924, 3432, 2100, 1672, 1534, 1534, 1672, 2100, 3432, 12870, 7788, 6118, 5506, 5341, 5506, 6118, 7788, 12870, 48620, 29172, 22692, 20152, 19174

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A033114. Diagonal sums are A115256. T(2n,n) is A115257. Corresponds to the triangle of antidiagonals of the correlation matrix of the sequence array for C(2n,n).

Let s=(1,2,6,20,...), (central binomial coefficients), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A115255 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203005 for characteristic polynomials of principal submatrices of M, with interlacing zeros. - Clark Kimberling, Dec 27 2011

			Triangle begins:
  1;
  2, 2;
  6, 5, 6;
  20, 14, 14, 20;
  70, 46, 41, 46, 70;
  252, 160, 134, 134, 160, 252;
Northwest corner (square format):
  1    2    6    20    70
  2    5    14   46    160
  6    14   41   134   466
  20   46   134  441   1534

Cf. A203004, A203001, A202453.

s[k_] := Binomial[2 k - 2, k - 1];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]]; (* A115255 in square format *)
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A006134 *)
Table[m[1, j], {j, 1, 12}]     (* A000984 *)
Table[m[j, j], {j, 1, 12}]     (* A115257 *)
Table[m[j, j + 1], {j, 1, 12}] (* 2*A082578 *)
(* Clark Kimberling, Dec 27 2011 *)

G.f.: 1/(sqrt(1-4*x)*sqrt(1-4*x*y)*(1-x^2*y)) (format due to Christian G. Bower).

T(n, k) = Sum_{j=0..n} [j<=k]*C(2*k-2*j, k-j)*[j<=n-k]*C(2*n-2*k-2*j, n-k-j).

A134759 a(n) = 2*A000984(n) - (n+1).

Original entry on oeis.org

1, 2, 9, 36, 135, 498, 1841, 6856, 25731, 97230, 369501, 1410852, 5408299, 20801186, 80233185, 310235024, 1202160763, 4667212422, 18150270581, 70690527580, 275693057619, 1076515748858, 4208197927417, 16466861455176, 64495207366175, 252821212875478

Offset: 0

Gary W. Adamson, Nov 09 2007

nonn

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

Cf. A000108, A000984, A134758, A134760, A134761, A134762, A134763, A134770, A134771.

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

Table[2 Binomial[2n,n]-n-1,{n,0,30}] (* Harvey P. Dale, Aug 07 2023 *)

[2*binomial(2*n,n) -(n+1) for n in range(41)] # G. C. Greubel, May 28 2024

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

More terms from Harvey P. Dale, Aug 07 2023

A054441 Convolution of (shifted) A026671 with A000984 (central binomial coefficients of even order).

Original entry on oeis.org

0, 1, 5, 23, 103, 455, 1993, 8679, 37633, 162643, 701075, 3015563, 12948083, 55513327, 237705547, 1016736115, 4344766607, 18550920063, 79149527249, 337482635279, 1438155203665, 6125448713739, 26077796587441, 110974892937943, 472081467302933, 2007534192877275, 8534465842495133

Offset: 0

Wolfdieter Lang, Mar 21 2000

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

Cf. A000045, A000984, A026671.

List([0..30], n-> Sum([0..n], k-> Binomial(2*n, n-k)*Fibonacci(k) )); # G. C. Greubel, Jul 15 2019

[(&+[Binomial(2*n, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 15 2019

Table[SeriesCoefficient[x/((-x+Sqrt[1-4*x])*Sqrt[1-4*x]),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 09 2012 *)

a(n):=sum(fib(k)*binomial(2*n,n-k),k,1,n); /* Vladimir Kruchinin, Mar 19 2016 */

x='x+O('x^66); concat([0],Vec(x/((-x+sqrt(1-4*x))*sqrt(1-4*x)))) \\ Joerg Arndt, May 06 2013

[sum(binomial(2*n, n-k)*fibonacci(k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jul 15 2019

G.f.: cbie(x)*x/(-x+1/cbie(x)), with cbie(x)=1/sqrt(1-4*x) = g.f. for A000984.
a(n) = Sum_{k=0..n} A026671(k-1)*binomial(2*(n-k), n-k), with A026671(-1):= 0.
a(n) = A026671(n) - binomial(2*n, n).
a(n) = Sum_{k=1..n} a(k-1)*binomial(2*(n-k), n-k) + 4^(n-1), n >= 1.
Recurrence: (n-2)*a(n) = 2*(4*n-9)*a(n-1) - (15*n-38)*a(n-2) - 2*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 09 2012
a(n) ~ (sqrt(5)+2)^n/sqrt(5). - Vaclav Kotesovec, Oct 09 2012
a(n) = Sum_{k=1..n} binomial(2*n,n-k)*F(k), where F denotes a Fibonacci number (A000045). - Vladimir Kruchinin, Mar 19 2016

A080397 Largest squarefree number dividing central binomial coefficient A000984(n).

Original entry on oeis.org

1, 2, 6, 10, 70, 42, 462, 858, 4290, 24310, 92378, 176358, 1352078, 520030, 222870, 6463230, 200360130, 129644790, 907513530, 1767263190, 22974421470, 134564468610, 526024740930, 22870640910, 1074920122770, 1504888171878, 1967930686302, 34766775458002, 1912172650190110

Offset: 0

Labos Elemer, Mar 19 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A007947, A000984, A001405, A048633.

a := n -> convert(numtheory:-factorset(binomial(2*n, n)), `*`):
seq(a(n),n=0..25); # Peter Luschny, Oct 31 2015

a[n_] := Times @@ FactorInteger[Binomial[2n, n]][[All, 1]]; Array[a, 26, 0] (* Jean-François Alcover, Jun 04 2019 *)

a(n) = vecprod(factor(binomial(2*n, n))[, 1]); \\ Amiram Eldar, Jun 21 2024

a(n) = A007947(A000984(n)).

More terms from Amiram Eldar, Jun 21 2024

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A157004 Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.

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A038665 Convolution of A007054 (super ballot numbers) with A000984 (central binomial coefficients).

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A053214 Central binomial coefficients (A000984) read mod 2n, with a(0)=1.

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A134758 a(n) = A000984(n) + n.

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A134770 a(n) = 4*A000984(n) - 3.

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A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

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A115255 "Correlation triangle" of central binomial coefficients A000984.

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