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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A134764 A054525 * A000984.

Original entry on oeis.org

1, 1, 5, 18, 69, 245, 923, 3412, 12864, 48549, 184755, 705162, 2704155, 10399675, 40116525, 155114088, 601080389, 2333593104, 9075135299, 35345215162, 137846527891, 538257689683, 2104098963719, 8233430018756, 32247603683030
Offset: 1

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Examples

			a(4) = 18 = (0, -1, 0, 1) dot (1, 2, 6, 20), where A000984 = (1, 2, 6, 20, 70, 252, ...) and (0, -1, 0, 1) = row 4 of triangle A054525.

Crossrefs

Cf. A000984, A054525.

Programs

  • Maple
    read("transforms") : A000984 := proc(n) binomial(2*n,n) ; end: a000984 := [seq(A000984(n),n=0..50)] ; a134764 := MOBIUS(a000984) ; for i from 1 to nops(a134764) do printf("%d,",op(i,a134764)) ; od: # R. J. Mathar, Jan 19 2009

Formula

Möbius transform of A000984.

Extensions

More terms from R. J. Mathar, Jan 19 2009

A145890 Triangle read by rows: T(n,k) = B(k)C(n-k), where B(j) is the central binomial coefficient binomial(2j,j) (A000984) and C(j) is the Catalan number binomial(2j,j)/(j+1) (A000108); 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 2, 6, 5, 4, 6, 20, 14, 10, 12, 20, 70, 42, 28, 30, 40, 70, 252, 132, 84, 84, 100, 140, 252, 924, 429, 264, 252, 280, 350, 504, 924, 3432, 1430, 858, 792, 840, 980, 1260, 1848, 3432, 12870, 4862, 2860, 2574, 2640, 2940, 3528, 4620, 6864, 12870, 48620
Offset: 0

Views

Author

Emeric Deutsch, Nov 22 2008

Keywords

Comments

Sum of entries in row n is (1/2)binomial(2n+2,n+1) = A001700(n).
T(n,0) = binomial(2n,n)/(n+1) = A000108(n) (the Catalan numbers).
T(n,n) = binomial(2n,n) = A000984(n) (the central binomial coefficients).

Examples

			Triangle starts:
   1;
   1,  2;
   2,  2,  6;
   5,  4,  6, 20;
  14, 10, 12, 20, 70;

Crossrefs

Cf. A000108, A000984, A001700.

Programs

  • Maple
    b:=proc(n) options operator, arrow: binomial(2*n,n) end proc: c:=proc(n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: T:=proc(n,k) if k <= n then b(k)*c(n-k) else 0 end if end proc: for n from 0 to 8 do seq(T(n,k), k =0..n) end do; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] /; 0 <= k <= n := Binomial[2k, k]*Binomial[2n - 2k, n-k]/(n-k+1);
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 23 2024 *)

Formula

T(n,k) = binomial(2k,k)*binomial(2n-2k,n-k)/(n-k+1) (0 <= k <= n).
G.f. = G(t,x) = (1-sqrt(1-4x))/(2x*sqrt(1-4tx)).

A157163 Product_{n>=1} (1 + 2*a(n)*x^n) = Sum_{k>=0} binomial(2*k, k)*x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).

Original entry on oeis.org

1, 3, 4, 27, 48, 156, 576, 2955, 7168, 27792, 95232, 352188, 1290240, 5105856, 17743872, 77010795, 252641280, 1000224768, 3616800768, 14484040464, 52102692864, 208963943424, 764877471744, 3025006038012, 11258183024640, 44968060784640, 166308918329344
Offset: 1

Views

Author

Wolfdieter Lang, Aug 10 2009

Keywords

Comments

In the original problem 2*a(n) = [2, 6, 8, 54, 96, 312, 1152, 5910, 14336, 55584, 190464, 704376, ...] appears.

Examples

			Recurrence I: a(4) = binomial(8, 4)/2 - 2*a(1)*a(3) = 35 - 8 = 27.
Recurrence II: a(4) = (1/2)*(1/2)*(-2*a(2))^2 + (1/2)*(1*cbi(4) - (1/2)*(2*cbi(1)*cbi(3) + 1*cbi(2)^2) + (1/3)*3*cbi(1)^2*cbi(2)) = 27.
Recurrence II (rewritten): a(4)= (1/8)*((-2)^4 + 2*(-2*a(2))^2 + (1/2)*4^4) = 27.

Links

Crossrefs

Cf. A147542 (Fibonacci), A157161 (Catalan).

Programs

  • Maple
    N:= 100: # for a(1)..a(N)
S:= convert(series(-ln(1-4*x)/2,x,N+1),polynom):
for n from 1 to N do
  a[n]:= coeff(S,x,n)/2;
  S:= S - add((-1)^(k-1)*(2*a[n])^k*x^(k*n)/k, k=1..N/n)
od:
seq(a[n],n=1..N); # Robert Israel, Jan 03 2019
  • PARI
    a(n) = if (n==1, 1, (1/(2*n))*((-2*a(1))^n + sumdiv(n, d, if ((d!=1) && (d!=n), d*(-2*a(d))^(n/d), 0)) + 4^n/2)); \\ after 2nd Recurrence II; Michel Marcus, Jul 06 2015

Formula

Recurrence I: With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions fp):
a(n) = binomial(2*n, n)/2 - Sum_{m=2..maxm(n)} 2^(m-1)*(Sum_{fp from FP(n,m)} (Product_{j=1..m} a(k[j]))), with maxm(n) = A003056(n) and the distinct parts k[j], j = 1..m, of the partition fp of n, n >= 3. Inputs a(1) = 1, a(2) = 3. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: With P(n,m) the set of all partitions of n with m parts, and the multinomial numbers M0 (given for every partition under A048996):
a(n) = (1/2)*Sum_{d|n, 1= 2; a(1) = 1, with cbi(n) = binomial(2*n, n) = A000984(n). The exponents e(j) >= 0 satisfy Sum_{j=1..n} j*e(j) =n and Sum_{j=1..n} e(j) = m. The M0 numbers are m!/(Product_{j=1..n} (e(j))!).
Recurrence II (rewritten, due to email from V. Jovovic, Mar 10 2009):
a(n) = ((-2*a(1))^n + Sum_{d|n, 1

A167481 Convolution of the central binomial coefficients A000984(n) and (-2)^n.

Original entry on oeis.org

1, 0, 6, 8, 54, 144, 636, 2160, 8550, 31520, 121716, 462000, 1780156, 6840288, 26436024, 102245472, 396589446, 1540427328, 5994280644, 23356702512, 91133123796, 355991626848, 1392115710024, 5449199307552, 21349205067996
Offset: 0

Views

Author

Paul Barry, Nov 04 2009

Keywords

Comments

Hankel transform is A102591.

Links

Crossrefs

Cf. A000984, A102591.

Programs

  • Mathematica
    Table[FullSimplify[(-2)^n/Sqrt[3] + 1/2*Binomial[2*(1+n),1+n] * Hypergeometric2F1[1,3/2+n,2+n,-2]],{n,0,20}] (* Vaclav Kotesovec, Jan 31 2014 *)
CoefficientList[Series[1/((1 + 2*t)*Sqrt[1 - 4 t]), {t,0,50}], t] (* G. C. Greubel, Jun 13 2016 *)

Formula

G.f.: 1/((1+2x)*sqrt(1-4x)).
a(n) = Sum_{k=0..n} (-2)^(n-k)*C(2k,k).
Conjecture: n*a(n) + 2*(1-n)*a(n-1) + 4*(1-2n)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011
a(n) = (-2)^n*JacobiP(n, 1/2, -1-n, -5). - Peter Luschny, Aug 02 2014

A171661 Triangle T(n,k) (n >= 0, 0 <= k <= n) read by rows: T(n,0) = T(n,1) = A000984(n); for n >= 2 and k >= 2, T(n,k) = T(n,k-1) - T(n-1,k-2).

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 20, 20, 14, 8, 70, 70, 50, 30, 16, 252, 252, 182, 112, 62, 32, 924, 924, 672, 420, 238, 126, 64, 3432, 3432, 2508, 1584, 912, 492, 254, 128, 12870, 12870, 9438, 6006, 3498, 1914, 1002, 510, 256, 48620, 48620, 35750, 22880, 13442, 7436
Offset: 0

Views

Author

Mark Dols, Dec 14 2009

Keywords

Examples

			Triangle starts:
1
2,2
6,6,4
20,20,14,8
70,70,50,30,16

Crossrefs

Cf. A000984, A172021.

Extensions

Edited by N. J. A. Sloane, Jan 24 2010
More terms from Mark Dols, Jan 24 2010

A199816 G.f.: exp( Sum_{n>=1} A000984(n)*A000172(n)/4 * x^n/n ), which involves central binomial coefficients (A000984) and Franel numbers (A000172).

Original entry on oeis.org

1, 1, 8, 101, 1639, 30665, 630225, 13836981, 319062453, 7640441894, 188534274850, 4767113222750, 122998902095908, 3228067183537455, 85960229675478804, 2317956019913480326, 63193008693741620771, 1739473925024629613227, 48292271242981605779173
Offset: 0

Views

Author

Paul D. Hanna, Nov 11 2011

Keywords

Comments

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.
Compare to the g.f. of the Catalan numbers (A000108): exp(Sum_{n>=1} A000984(n)/2*x^n/n) and to the g.f. of A166991: exp(Sum_{n>=1} A000172(n)/2*x^n/n).

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 101*x^3 + 1639*x^4 + 30665*x^5 +...
where
log(A(x)) = 1*1*x + 3*5*x^2/2 + 10*28*x^3/3 + 35*173*x^4/4 + 126*1126*x^5/5 + 462*7592*x^6/6 +...+ A000984(n)/2*A000172(n)/2*x^n/n +...

Crossrefs

Cf. A199813, A166991, A000108, A181418, A000984, A000172.

Programs

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

Formula

Convolution 4th power yields A199813.

A203194 (n-1)-st elementary symmetric function of the first n terms of (1,2,6,20,70,252,...)=A000984.

Original entry on oeis.org

1, 3, 20, 412, 29080, 7344960, 6790976640, 23310543674880, 300020122552550400, 14587151144134593024000, 2695072097623041659787264000, 1901191652075515716657381408768000, 5141119908014521906432306538772430848000
Offset: 1

Views

Author

Clark Kimberling, Dec 30 2011

Keywords

Crossrefs

Cf. A000984.

Programs

  • Mathematica
    f[k_] := Binomial[2 k - 2, k - 1];
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 14}]  (* A203194 *)

A239226 a(n) = A000984(n) * A081085(n).

Original entry on oeis.org

1, 8, 120, 2240, 47320, 1084608, 26330304, 666631680, 17419647960, 466416716480, 12730856057920, 352914423912960, 9908504597118400, 281166914888384000, 8050729214434752000, 232310201739468042240, 6748710905805484610520, 197211871554285957969600
Offset: 0

Views

Author

Michael Somos, Mar 12 2014

Keywords

Comments

Denoted s_4B by Piezas.

Examples

			G.f. = 1 + 8*x + 120*x^2 + 2240*x^3 + 47320*x^4 + 1084608*x^5 + 26330304*x^6 + ...

Links

Crossrefs

Cf. A000984, A081085.

Programs

  • Magma
    [Binomial(2*n,n)*(&+[Binomial(n, k)*Binomial(2*k, k)*Binomial(2*n - 2*k, n - k): k in [0..n]]): n in [0..50]]; // G. C. Greubel, Aug 07 2018
  • Mathematica
    Table[Binomial[2*n, n]*Sum[Binomial[n, k]*Binomial[2*k, k]*Binomial[2*n - 2*k, n - k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 07 2018 *)
  • PARI
    {a(n) = binomial(2*n, n) * sum(k=0, n, binomial(n, k) * binomial(2*k, k) * binomial(2*n - 2*k, n-k))};

Formula

D-finite with recurrence 0 = a(n) * n^3 - a(n-1) * 8 * (2*n - 1) * (3*n^2 - 3*n + 1) + a(n-2) * 128 * (n-1) * (2*n - 1) * (2*n - 3) for all n in Z.

A277339 Exponential self-convolution of this sequence gives central binomial coefficients (A000984).

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 26, 92, 64, -1328, 2272, 86912, -157706, -7271042, 17815604, 853696664, -2615703541, -133125019397, 490820087366, 26636670621548, -114924854384183, -6653655394184683, 32904766004185814, 2029701686588972068, -11322597283993315976
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 09 2016

Keywords

Links

Crossrefs

Cf. A000984.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, (
      binomial(2*n, n)-add(a(k)*a(n-k)*
      binomial(n, k), k=1..n-1))/2)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 12 2016
  • Mathematica
    Table[SeriesCoefficient[Exp[x] Sqrt[BesselI[0, 2 x]], {x, 0, n}] n!, {n, 0, 25}]
  • PARI
    x = 'x + O('x^30); serlaplace(exp(x)*sqrt(besseli(0, 2*x))) \\ Michel Marcus, Oct 09 2016

Formula

E.g.f.: exp(x)*sqrt(BesselI_0(2*x)).

A329533 First differences of A051924, or second differences of Central binomial coefficients A000984.

Original entry on oeis.org

3, 10, 36, 132, 490, 1836, 6930, 26312, 100386, 384540, 1478048, 5697720, 22019556, 85284920, 330961950, 1286562960, 5009003250, 19528599420, 76231136520, 297910080600, 1165429743660, 4563490674600, 17884841191620, 70148829799152, 275344923755700, 1081512966189656, 4250730282412320
Offset: 0

Views

Author

M. F. Hasler, Nov 15 2019

Keywords

Crossrefs

Cf. A000984, A051924, A163771, A094796.

Programs

  • Mathematica
    Differences[#, 2] &@ Array[Binomial[2 #, #] &, 29, 0] (* Michael De Vlieger, Nov 15 2019 *)
  • PARI
    C=vector(30,n,binomial(2*n--,n));C=C[^1]-C[^-1];C=C[^1]-C[^-1]

Formula

a(n) = A051924(n) - A051924(n-1) = A000984(n+2) - 2*A000984(n+1) + A000984(n).
a(n) = 3*(3*n+2)*(n+1)*binomial(2*n+4,n+2)/(4*(2*n+1)*(2*n+3)). - Alois P. Heinz, Sep 13 2024
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