cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A387086 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.

Original entry on oeis.org

1, 0, 2, 4, 16, 52, 188, 672, 2458, 9052, 33648, 125864, 473500, 1789632, 6791528, 25863568, 98796096, 378411332, 1452886052, 5590262688, 21551271916, 83228809640, 321933018272, 1247062996304, 4837152438556, 18785529571200, 73037938668632, 284268423472432
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2025

Keywords

Crossrefs

Cf. A183161, A208977, A387084.
Cf. A000108, A000984, A387085.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] - 1)], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(4*x-1+2*sqrt(1-4*x)))

Formula

Sum_{k=0..n} a(k) * a(n-k) = A387085(n).
G.f.: 1/sqrt( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/sqrt(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g/sqrt((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
a(n) ~ 2^(2*n - 1/2) / (Gamma(1/4) * n^(3/4)) * (1 - Gamma(1/4)^2/(16*Pi*sqrt(2*n))). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*(n-1)*a(n) -2*(n-1)*(10*n-17)*a(n-1) +4*(4*n^2-24*n+29)*a(n-2) +32*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

A038602 One half of convolution of central binomial coefficients A000984(n) with A000984(n+2), n >= 0.

Original entry on oeis.org

3, 16, 73, 316, 1334, 5552, 22901, 93892, 383290, 1559680, 6331098, 25649976, 103758828, 419195552, 1691825933, 6822051092, 27488564498, 110691186272, 445487285678, 1792047789512, 7205785665908, 28963557761312
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A000346 with Catalan numbers but with C(0)=1 replaced by 3

Links

Crossrefs

Cf. A000108, A000346.

Programs

  • Mathematica
    CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x)*((1-Sqrt[1-4*x])/(2*x)+2)/(1-4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 28 2014 *)

Formula

a(n) = 2^(2*n+3)-(3*n+5)*C(n+1), C(n): Catalan numbers A000108.
G.f.: c(x)*(c(x)+2)/(1-4*x), where c(x) is G.f. for Catalan numbers.
a(n) ~ 2^(2*n+3) * (1-3/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Mar 28 2014
Recurrence: n*(n+2)*a(n) = 2*(4*n^2 + 5*n - 1)*a(n-1) - 8*(n+1)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Mar 28 2014

A038697 Convolution of A000917 with A000984 (central binomial coefficients).

Original entry on oeis.org

3, 26, 163, 894, 4558, 22196, 104739, 483062, 2189530, 9789900, 43295118, 189749676, 825364668, 3567219688, 15332925731, 65591312550, 279415474594, 1185903736412, 5016725589402, 21159849864964, 89012979703940
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A007054 (Super ballot numbers) with A002697;

Links

Crossrefs

Cf. A007054, A038665, A038679, A000917, A000108, A000984, A002697.

Programs

  • Maple
    seq(n*4^(n+1)+binomial(2*n+3,n+1),n=0..30); # Robert Israel, May 22 2019

Formula

a(n) = n*4^(n+1)+binomial(2*n+3, n+1).
G.f.: c(x)*(4-c(x))/(1-4*x)^2, where c(x) = g.f. for Catalan numbers A000108.
(160+64*n)*a(n) - (160+48*n)*a(n+1) + (50+12*n)*a(n+2) - (5+n)*a(n+3)=0. - Robert Israel, May 22 2019

A080392 Numbers k such that A000984(k) mod k = 0 and A080383(k) != 7.

Original entry on oeis.org

2, 420, 920, 1122, 1218, 1892, 1978, 2444, 2914, 3198, 3782, 4028, 4136, 4292, 4664, 4958, 4960, 5330, 5762, 5986, 6020, 6032, 6710, 6834, 6864, 6882, 6954, 6956, 6968, 7106, 7130, 7140, 7238, 7254, 7448, 7616, 8178, 8190, 8400, 8692, 9462, 9506, 10712, 11060, 11288
Offset: 1

Views

Author

Labos Elemer, Mar 17 2003

Keywords

Comments

Numbers arising in A067348 and not present in A080385.
Even numbers n such that n divides binomial(n, [n/2]) and A010551(n) does not divide j!*(n-j)! exactly 7 times for j = 0..n. - Peter Luschny, Aug 04 2017

Examples

			A080383(2) = 3;
A080383(420) = 11;
A080383(920) = 11;
A080383(1122) = 9;
A080383(1218) = 9.

Links

Crossrefs

Cf. A000984, A001405, A067348, A080383, A080385.

Programs

  • Maple
    isa := proc(n)  local bn, bm;
if n mod 2 = 0 then bn := binomial(n, iquo(n,2)):
if modp(bn, n) = 0 then
   bm := (n, j) -> `if`(modp(bn, binomial(n, j)) = 0, 1, 0):
   return 1 <> add(bm(n, j), j=2..iquo(n,2)-1)
fi fi; false end:
select(isa, [$1..5000]); # Peter Luschny, Aug 04 2017
  • Mathematica
    Do[s=Count[Table[IntegerQ[Binomial[n, Floor[n/2]]/ Binomial[n, j]], {j, 0, n}], True]; s1=IntegerQ[Binomial[n, n/2]/n]; If[ !Equal[s, 7] && Equal[s1, True], Print[n]], {n, 1, 10000}]
(* Second program: *)
Select[Range@ 5000, Function[n, And[Divisible[Binomial[n, n/2], n], Count[Table[Divisible[Binomial[n, Floor[n/2]], Binomial[n, j]], {j, 0, n}], True] != 7]]] (* Michael De Vlieger, Jul 30 2017 *)

Extensions

More terms from Michael De Vlieger, Jul 30 2017

A080395 Even numbers k such that the central binomial coefficient A000984(k, k/2) is divisible by k^2.

Original entry on oeis.org

1848, 2574, 4004, 4290, 6732, 7480, 8398, 12012, 12236, 17710, 20930, 22770, 24570, 24650, 24882, 25080, 25194, 26796, 27132, 30160, 31668, 36540, 36708, 37674, 37944, 38454, 47124, 47740, 51282, 51480, 53200, 57288, 62160, 68376, 69930, 70840, 73260, 75480, 83640
Offset: 1

Views

Author

Labos Elemer, Mar 18 2003

Keywords

Comments

a(n)/2 is a term of A121943 for all n. - Amiram Eldar, Mar 07 2022

Links

Crossrefs

Cf. A001405, A000984, A067348, A080394, A121943.

Programs

  • Mathematica
    Do[s=Binomial[n, n/2]/n^2; If[IntegerQ[s], Print[n]], {n, 1, 50000}]
Select[2Range[50000],Mod[Binomial[#,#/2],#^2]==0&] (* Harvey P. Dale, Jan 27 2025 *)

Extensions

Name corrected and more terms added by Amiram Eldar, Mar 07 2022

A082760 Trinomial transform of central binomial coefficients (A000984).

Original entry on oeis.org

1, 9, 133, 2283, 41589, 782149, 15007831, 291987327, 5738781717, 113670432141, 2265365597553, 45371770152927, 912463490703879, 18413786266130979, 372689501160357787, 7562267679664012693, 153785739808245476757
Offset: 0

Views

Author

Emanuele Munarini, May 21 2003

Keywords

Formula

a(n) = Sum[ Binomial[2k, k] Trinomial[n, k], {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907)
a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)C(2(j+k), j+k)}}; - Paul Barry, Feb 15 2005

A103821 A Whitney transform of the central binomial coefficients A000984.

Original entry on oeis.org

1, 3, 11, 43, 179, 771, 3395, 15171, 68515, 311907, 1428835, 6578531, 30414435, 141105251, 656588899, 3063038051, 14321092195, 67088405091, 314825048675, 1479654425187, 6963888239203, 32815960756835, 154813864252003
Offset: 0

Views

Author

Paul Barry, Feb 16 2005

Keywords

Comments

Partial sums of A006139. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)).

Links

Programs

  • Mathematica
    CoefficientList[Series[1/((1-x)*Sqrt[1-4*x-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

Formula

G.f. : 1/((1-x)sqrt(1-4x-4x^2));
a(n)=sum{k=0..n, sum{i=0..n, C(k, i-k)}*C(2k, k)}.
Conjecture: n*a(n) +(2-5n)*a(n-1) +2*a(n-2)+4*(n-1)*a(n-3)=0. - R. J. Mathar, Dec 14 2011
a(n) ~ sqrt(34+23*sqrt(2))*(2+2*sqrt(2))^n/(7*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012

A120278 a(n) = Sum_{m=1..n} Sum_{k=1..m} C(2*k,k), where C(2*k,k) = (2*k)!/(k!)^2 = A000984(k).

Original entry on oeis.org

2, 10, 38, 136, 486, 1760, 6466, 24042, 90238, 341190, 1297574, 4958114, 19019254, 73196994, 282492254, 1092867904, 4236849774, 16455966944, 64020347914, 249431257704, 973100041934, 3800867789884, 14862066265434, 58170868424084
Offset: 1

Views

Author

Alexander Adamchuk, Jul 04 2006

Keywords

Comments

a(2*(p-1)) is divisible by p^2 for p=7,13,19,31,37,43,61,67.. A002476 (Primes of the form 6m + 1).

Links

Crossrefs

Cf. A000984, A066796, A002476.
Cf. A066796, A079309.

Programs

  • Mathematica
    Table[Sum[Sum[(2k)!/(k!)^2,{k,1,m}],{m,1,n}],{n,1,50}]
CoefficientList[Series[(1/Sqrt[1-4 x]-1)/((x-1)^2 x),{x,0,50}],x] (* Harvey P. Dale, May 24 2011 *)

Formula

a(n) = Sum_{m=1..n} Sum_{k=1..m} (2*k)!/(k!)^2.
a(n) = 2 * Sum_{k=1..n} A079309(k) = Sum_{k=1..n} A066796(k). - Alexander Adamchuk, Sep 01 2006
G.f.: x*(1/sqrt(1-4*x)-1)/(x*(x-1)^2). - Harvey P. Dale, May 24 2011
Recurrence: n*a(n) = 2*(3*n-1)*a(n-1) - (9*n-4)*a(n-2) + 2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+4)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

A128079 a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1), where A000984 is the central binomial coefficients and A001263 is the Narayana triangle.

Original entry on oeis.org

1, 3, 13, 69, 411, 2633, 17739, 124029, 892327, 6567285, 49235715, 374841195, 2890994445, 22545855855, 177524073021, 1409591810133, 11275693221519, 90792020672429, 735367765159347, 5987665336600683, 48987680485918149
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2007

Keywords

Examples

			Illustrate a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1) by:
a(2) = 1*(1) + 2*(3) + 6*(1) = 13;
a(3) = 1*(1) + 2*(6) + 6*(6) + 20*(1) = 69;
a(4) = 1*(1) + 2*(10)+ 6*(20)+ 20*(10)+ 70*(1) = 411.
The Narayana triangle A001263(n+1,k+1) = C(n,k)*C(n+1,k)/(k+1) begins:
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 10, 20, 10, 1;
1, 15, 50, 50, 15, 1; ...

Links

Crossrefs

Cf. A000984, A001263.

Programs

  • Mathematica
    Table[Sum[Binomial[2*k,k]*Binomial[n,k]*Binomial[n+1,k]/(k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(2*k,k)*binomial(n,k)*binomial(n+1,k)/(k+1))}

Formula

a(n) = Sum_{k=0..n} C(2k,k)*C(n,k)*C(n+1,k)/(k+1).
Recurrence: (n+1)*(n+2)*a(n) = (7*n^2+11*n+6)*a(n-1) + 3*(7*n^2-19*n+6)*a(n-2) - 27*(n-2)*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3^(2*n+7/2)/(8*Pi*n^2) . - Vaclav Kotesovec, Oct 20 2012
a(n) = ((n+3)^2*A005802(n+1)-(n-3)*(n+1)*A005802(n))/12. - Mark van Hoeij, Nov 12 2023

A131764 Inverse Euler transform of central binomial coefficients A000984.

Original entry on oeis.org

1, 2, 3, 10, 30, 102, 335, 1170, 4080, 14560, 52377, 190650, 698870, 2581110, 9586395, 35791358, 134215680, 505290270, 1908866960, 7233629130, 27487764474, 104715392730, 399822314775, 1529755308210, 5864061663920, 22517998136832, 86607683851185, 333599972392960, 1286742745883790, 4969489243995030, 19215358392200893, 74382032555280450, 288230376084602880
Offset: 0

Views

Author

F. Chapoton, Oct 04 2007

Keywords

Comments

This is the sequence of dimensions of a free Lie algebra on some specific set of generators.

Examples

			2*x + 3*x^2 + 10*x^3 + 30*x^4 + 102*x^5 + 335*x^6 + 1170*x^7 + 4080*x^8 + ...
(1-x)^(-2)*(1-x^2)^(-3)*(1-x^3)^(-10)*(1-x^4)^(-30)*(1-x^5)^(-102) = 1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 + ... .

Links

Crossrefs

Cf. A022553, A000984, A000108.

Programs

  • Maple
    # The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> binomial(2*n, n)):
seq(a(n), n = 0..32); # Peter Luschny, Nov 21 2022
  • Mathematica
    a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*2^(2*#-1)&]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Feb 20 2017 *)
  • MuPAD
    a(n):=proc(n) begin 1/n*_plus(moebius(n/d)*2^(2*d-1)$d in divisors(n)) end;
  • PARI
    a(n)=sumdiv(n,d,1/n*moebius(n/d)*2^(d*2-1)); /* Joerg Arndt, Jul 06 2011 */
  • PARI
    {a(n) = local(A); if( n<1, 0, A = sqrt(1 - 4*x + x * O(x^n)); for( k=1, n-1, A *= (1 - x^k + x * O(x^n))^ polcoeff( A, k)); -polcoeff( A, n))} /* Michael Somos, Apr 01 2012 */

Formula

a(n) = (1/n) * Sum_{d|n} moebius(n/d)*2^(2*d-1) for n > 0, a(0) = 1.
a(n) ~ 2^(2*n-1) / n. - Vaclav Kotesovec, Oct 09 2019

Extensions

More explicit definition from Michael Somos, Apr 01 2012. - N. J. A. Sloane, Feb 20 2017
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