A000984 -id:A000984 - cp's OEIS frontend

A264960 Half-convolution of the central binomial coefficients A000984 with itself.

1, 2, 10, 32, 146, 512, 2248, 8192, 35218, 131072, 556040, 2097152, 8815496, 33554432, 140107040, 536870912, 2230302098, 8589934592, 35541690568, 137438953472, 566823203656, 2199023255552, 9044910175520, 35184372088832, 144393718191496

Offset: 0

Peter Bala, Nov 29 2015

The half-convolution of a sequence {s(n)}n>=0 with itself is defined by r(n) := Sum_{k = 0..floor(n/2)} s(k)*s(n-k). See A201204.

Muniru A Asiru, Table of n, a(n) for n = 0..1520

Cf. A002894, A013776, A112830.

List([0..24],n->Sum([0..Int(n/2)],k->Binomial(2*k,k)*Binomial(2*n-2*k,n-k))); # Muniru A Asiru, Nov 25 2018

[(&+[Binomial(2*k,k)*Binomial(2*n-2*k, n-k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Nov 26 2018

A264960:= n-> add(binomial(2*k,k)*binomial(2*n - 2*k, n - k),k = 0..floor(n/2)):
seq(A264960(n),n = 0..24);

a[n_] := Sum[Binomial[2k, k]*Binomial[2n - 2k, n - k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* Amiram Eldar, Nov 25 2018 *)

a(n) = sum(k = 0, n\2, binomial(2*k,k)*binomial(2*n - 2*k, n - k)); \\ Michel Marcus, Nov 30 2015

[sum(binomial(2*k,k)*binomial(2*n-2*k, n-k) for k in (0..floor(n/2))) for n in range(30)] # G. C. Greubel, Nov 26 2018

a(n) = Sum_{k = 0..floor(n/2)} binomial(2*k,k)*binomial(2*n - 2*k, n - k).
a(2*n + 1) = 2^(4*n + 1) = A013776(n).
a(2*n) = (1/2)*(binomial(2*n,n)^2 + 16^n) = A112830(2*n,n).
O.g.f.: (1/2)*( 2/Pi*EllipticK(4*x) + 1/(1 - 4*x) ).
E.g.f.: (1/2)*( cosh(4*x) + sinh(4*x) + (BesselI(0,2*x))^2 ).
D-finite with recurrence: - (2*n-3)*n^2*a(n) + 4*(2*n-1)*(n-1)^2*a(n-1) + 16*(2*n-3)*(n-1)^2*a(n-2) - 64*(2*n-1)*(n-2)^2*a(n-3) = 0. - Georg Fischer, Nov 25 2022

A081393 a(n) is the smallest k such that number of non-unitary prime divisors of central binomial coefficient, A000984(k) = C(2*k,k) equals n.

Original entry on oeis.org

1, 3, 5, 14, 48, 74, 182, 314, 480, 774, 960, 1321, 1323, 1670, 3121, 3455, 3457, 3472, 3462, 3469, 8203, 9991, 12163, 15838, 15840, 17665, 18480, 18482, 19458, 19464, 36782, 19865, 36789, 40048, 43603, 43655, 47518, 61654, 61653, 61685, 61684, 87120, 92958, 93181, 93185, 93187, 93191

Offset: 0

Labos Elemer, Mar 27 2003

nonn

			n=5: a(5)=74, C(148,74) has 5 non-unitary prime divisors: {2,3,5,7,11} and 74 is the smallest.

Cf. A081387, A081394, A081395.

seq[len_, kmax_] := Module[{s = Table[0, {len}], k = 1, c = 0, i}, While[c < len && k < kmax, i = Count[FactorInteger[Binomial[2*k, k]][[;; , 2]], ?(# > 1 &)] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = k]; k++]; TakeWhile[s, # > 0 &]]; seq[20, 10^4] (* _Amiram Eldar, May 15 2023 *)

a(n) = Min{k; A081387(k) = n}.

a(11)-a(21) from Michel Marcus, Sep 01 2019

a(22)-a(46) from Amiram Eldar, May 15 2023

A142962 Scaled convolution of (n^3)*A000984(n) with A000984(n).

Original entry on oeis.org

4, 26, 81, 184, 350, 594, 931, 1376, 1944, 2650, 3509, 4536, 5746, 7154, 8775, 10624, 12716, 15066, 17689, 20600, 23814, 27346, 31211, 35424, 40000, 44954, 50301, 56056, 62234, 68850, 75919, 83456, 91476, 99994, 109025, 118584, 128686, 139346, 150579

Offset: 1

Wolfdieter Lang, Sep 15 2008

S(3,n) := Sum_{j=0..n} j^3*binomial(2*j,j)*binomial(2*(n-j),n-j).
a(n) = 2^3*S(3,n)/4^n, n >= 1.
O.g.f. for S(3,n) is G(k=3,x). See triangle A142963 for the general G(k,x) formula.
The author was led to compute such sums by a question asked by M. Greiter, Jun 27 2008.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A142961 triangle: row k=3: [3, 5], with the row polynomial 3+5*n.

Cf. A049451 (scaled k=2 case).

Rest@ CoefficientList[Series[x (4 + 10 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, Jul 02 2023 *)

a(n) = n^2*(3+5*n)/2.
a(n) = (2^3)*S(3,n)/4^n with the convolution S(3,n) defined above.
G.f.: x*(4+10*x+x^2)/(1-x)^4. - Joerg Arndt, Jul 02 2023

A152229 Eigentriangle, row sums = A000984.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 9, 3, 2, 6, 29, 9, 6, 6, 20, 97, 29, 18, 18, 20, 70, 333, 97, 58, 54, 60, 70, 252, 1165, 333, 194, 174, 180, 210, 252, 924, 4135, 1165, 666, 582, 580, 630, 756, 924, 3432, 14845, 4135, 2330, 1998, 1940, 2030, 2268, 2772, 3432, 12870

Offset: 0

Gary W. Adamson, Nov 29 2008

Row sums = A000984: (1, 2, 6, 20, 70, 252,...), left border = A081696.

Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
3, 1, 2;
9, 3, 2, 6;
29, 9, 6, 6, 20;
97, 29, 18, 18, 20, 70;
333, 97, 58, 54, 60, 70, 252;
1165, 333, 194, 174, 180, 210, 252, 924;
4135, 1165, 666, 582, 580, 630, 756, 924, 3432;
14845, 4135, 2330, 1998, 1940, 2030, 2268, 2772, 3432, 12870;
...
Row 3 = (9, 3, 2, 6) = termwise products of (9, 3, 1, 1) and (1, 1, 2, 6).

Cf. A000984, A081696.

Triangle read by rows, M*Q. M = an infinite lower triangular matrix with A081696: (1, 1, 3, 9, 29, 97, 333, 1165,...) in every column; and Q = a matrix with A000984 as the main diagonal (prefaced with a 1): (1, 1, 2, 6, 20, 70, 252,...) and the rest zeros.

A199813 G.f.: exp( Sum_{n>=1} A000984(n)A000172(n) x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

Original entry on oeis.org

1, 4, 38, 504, 8249, 154036, 3149326, 68741880, 1576163328, 37548785408, 922252542128, 23222906277952, 596981991939677, 15616173859832740, 414621835401615110, 11150969618415168280, 303278916800906999191, 8330190277527648516572, 230814933905555392525290

Offset: 0

Paul D. Hanna, Nov 10 2011

nonn

Sum_{k=0..n} C(n,k)^2 = A000984(n) defines central binomial coefficients.

Sum_{k=0..n} C(n,k)^3 = A000172(n) defines Franel numbers.

			G.f.: A(x) = 1 + 4*x + 38*x^2 + 504*x^3 + 8249*x^4 + 154036*x^5 +...
where
log(A(x)) = 2*2*x + 6*10*x^2/2 + 20*56*x^3/3 + 70*346*x^4/4 + 252*2252*x^5/5 + 924*15184*x^6/6 +...+ A000984(n)*A000172(n)*x^n/n +...

Cf. A199816, A181418, A000984, A000172.

{a(n)=polcoeff(exp(sum(m=1,n,binomial(2*m, m)*sum(k=0, m, binomial(m, k)^3)*x^m/m)+x*O(x^n)),n)}

A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.

Original entry on oeis.org

1, 2, 18, 140, 1330, 12852, 130284, 1348776, 14247090, 152618180, 1654120468, 18096447096, 199536967084, 2214714164600, 24720932068200, 277289164574640, 3123590583844530, 35318969120870820, 400692715550057700, 4559427798654821400, 52020436064931914580

Offset: 0

Paul D. Hanna, Sep 08 2012

nonn

			L.g.f.: L(x) = 2*x + 18*x^2/2 + 140*x^3/3 + 1330*x^4/4 + 12852*x^5/5 + 130284*x^6/6 + ...
where
exp(L(x)) = 1 + 2*x + 11*x^2 + 66*x^3 + 485*x^4 + 3842*x^5 + 32712*x^6 + ... + A216585(n)*x^n/n + ...
The central trinomial coefficients (A002426) begin:
[1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, ...];
The central binomial coefficients (A000984) begin:
[1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, ...].

G. C. Greubel, Table of n, a(n) for n = 0..925

Cf. A216585, A151341, A168595, A002426, A000984.

Table[Binomial[2*n, n]*Sum[ Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Feb 27 2017 *)

{a(n) = polcoeff((1+x+x^2)^n,n) * polcoeff((1+2*x+x^2)^n,n)}

{a(n)=binomial(2*n,n)*sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k))}
for(n=0,21,print1(a(n),", "))

a(n) = binomial(2*n, n) * Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k).
Logarithmic derivative of A216585, after ignoring initial term a(0).
a(n) = [x^n*y^n] ( 1 + (x + y)^2 + (x + y)^4 )^n. - Peter Bala, Feb 17 2020
G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+4*x))/sqrt(1+4*x). - Mark van Hoeij, May 13 2025

A216585 G.f.: exp( Sum_{n>=1} A000984(n)A002426(n)x^n/n ), where A000984 is the central binomial coefficients and A002426 is the central trinomial coefficients.

Original entry on oeis.org

1, 2, 11, 66, 485, 3842, 32712, 291568, 2697610, 25679316, 250190125, 2484270622, 25062816127, 256275246582, 2650947762450, 27697861115740, 291943603838698, 3101066786857876, 33167191013319532, 356924515784037128, 3862299973917286526, 42003704374124712172

Offset: 0

Paul D. Hanna, Sep 09 2012

nonn

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 66*x^3 + 485*x^4 + 3842*x^5 + 32712*x^6 +...
such that
log(A(x)) = 2*1*x + 6*3*x^2/2 + 20*7*x^3/3 + 70*19*x^4/4 + 252*51*x^5/5 + 924*141*x^6/6 +...+ A000984(n)*A002426(n)*x^n/n +...

Cf. A216584, A216586, A002426, A000984.

{a(n)=polcoeff(exp(sum(m=1,n+1,binomial(2*m,m)*polcoeff((1+x+x^2)^m,m)*x^m/m+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

Logarithmic derivative yields A216584.

A227325 a(n) = A000272(n+1) * A000984(n).

Original entry on oeis.org

1, 2, 18, 320, 8750, 326592, 15529668, 899678208, 61556811030, 4862000000000, 435644983598396, 43678490079264768, 4846281282497517772, 589650705050503577600, 78074729079345703125000, 11177395284330167371038720, 1720546364146510165684599270

Offset: 0

Reinhard Zumkeller, Jul 07 2013

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A000272, A000984.

Central terms of triangle A088956.

Haskell
```
a227325 n = a000272 (n + 1) * a000984 n
```

a(n) = A088956(2*n, n) = (n+1)^(n-1) * C(2*n, n).

Typo in formula fixed by Zak Seidov, Jul 08 2013

A289800 p-INVERT of the central binomial coefficients (A000984), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 17, 75, 336, 1517, 6879, 31276, 142439, 649431, 2963266, 13528285, 61785007, 282257992, 1289734455, 5894167695, 26939918564, 123142940445, 562928407213, 2573477722376, 11765383864555, 53790586563231, 245933621620228, 1124446028551665, 5141224466008849

Offset: 0

Clark Kimberling, Aug 12 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A289780 for a guide to related sequences.

Cf. A000984, A289780.

z = 60; s = x/Sqrt[1 - 4 x]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000984 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289800 *)

A292441 Largest m such that m^2 divides A000984(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 2, 2, 3, 2, 2, 2, 2, 10, 30, 12, 3, 6, 10, 10, 6, 2, 2, 60, 30, 42, 42, 28, 2, 4, 4, 4, 21, 14, 14, 6, 2, 2, 10, 140, 14, 126, 6, 60, 90, 12, 84, 84, 210, 30, 18, 12, 6, 36, 4, 4, 6, 4, 4, 12, 12, 132, 132, 440, 55, 330, 10, 10, 90, 30, 30, 180

Offset: 0

Seiichi Manyama, Sep 16 2017

nonn

a(n) is the product of p^floor(m(n,p)/2) over primes pRobert Israel, Sep 17 2017
Granville and Ramaré show that A006530(a(n)) > sqrt(n/5) if n >= 2082.
In particular a(n) -> infinity as n -> infinity. - Robert Israel, Sep 18 2017

			binomial(10,5)/7           =  252/7   = 36 = a(5)^2.
binomial(12,6)/(3*7*11)    =  924/231 =  4 = a(6)^2.
binomial(14,7)/(2*3*11*13) = 3432/858 =  4 = a(7)^2.

Robert Israel, Table of n, a(n) for n = 0..10000
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [DOI].
Eric Weisstein's World of Mathematics, Erdős Squarefree Conjecture
Wikipedia, Kummer's theorem

Cf. A000188, A000984, A006530, A292442.

A000188:= n -> mul(t[1]^floor(t[2]/2), t = ifactors(n)[2]):
seq(A000188(binomial(2*n,n)),n=0..100); # Robert Israel, Sep 17 2017

a(n) > 1 for n > 4.

a(n) = A000188(A000984(n)). - Robert Israel, Sep 17 2017

cp's OEIS Frontend

A264960 Half-convolution of the central binomial coefficients A000984 with itself.

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A081393 a(n) is the smallest k such that number of non-unitary prime divisors of central binomial coefficient, A000984(k) = C(2*k,k) equals n.

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A142962 Scaled convolution of (n^3)*A000984(n) with A000984(n).

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A152229 Eigentriangle, row sums = A000984.

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A199813 G.f.: exp( Sum_{n>=1} A000984(n)*A000172(n) * x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

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A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.

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A216585 G.f.: exp( Sum_{n>=1} A000984(n)*A002426(n)*x^n/n ), where A000984 is the central binomial coefficients and A002426 is the central trinomial coefficients.

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A227325 a(n) = A000272(n+1) * A000984(n).

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A199813 G.f.: exp( Sum_{n>=1} A000984(n)A000172(n) x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

A216585 G.f.: exp( Sum_{n>=1} A000984(n)A002426(n)x^n/n ), where A000984 is the central binomial coefficients and A002426 is the central trinomial coefficients.