A002458 -id:A002458 - cp's OEIS frontend

A100033 Bisection of A001700.

3, 35, 462, 6435, 92378, 1352078, 20058300, 300540195, 4537567650, 68923264410, 1052049481860, 16123801841550, 247959266474052, 3824345300380220, 59132290782430712, 916312070471295267, 14226520737620288370

Offset: 0

N. J. A. Sloane, Nov 20 2004

Cf. A001700, A002458, A187364, A187365.

a:=n->binomial(4*n+3,2*n+2): seq(a(n),n=0..19);

a(n) = binomial(4*n+3, 2*n+2). - Emeric Deutsch, Dec 09 2004
From Peter Bala, Mar 19 2023: (Start)
a(n) = (1/2)*Sum_{k = 0..2*n+2} binomial(2*n+2,k)^2.
a(n) = (1/2)*hypergeom([-2 - 2*n, -2 - 2*n], [1], 1).
a(n) = 2*(4*n + 1)*(4*n + 3)/((n + 1)*(2*n + 1)) * a(n-1). (End)
From Peter Bala, Mar 28 2023: (Start)
a(n) = (1/(2*n + 2))*Sum_{k = 0..2*n+2} k*binomial(2*n+2,k)^2.
a(n) = 2*(n + 1)*hypergeom([-1 - 2*n, -1 - 2*n], [2], 1). (End)

More terms from Emeric Deutsch, Dec 09 2004

A110145 a(n) = Sum_{k=0..n} C(n,k)^2*mod(k,2).

Original entry on oeis.org

0, 1, 4, 10, 32, 126, 472, 1716, 6400, 24310, 92504, 352716, 1351616, 5200300, 20060016, 77558760, 300533760, 1166803110, 4537591960, 17672631900, 68923172032, 269128937220, 1052049834576, 4116715363800, 16123800489472, 63205303218876, 247959271674352

Offset: 0

Paul Barry, Jul 13 2005

Interleaves A002458 and A037964.

Number of n-element subsets of [2n] having an odd sum. - Alois P. Heinz, Feb 06 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A159916.

Table[Sum[Binomial[n,k]^2 Mod[k,2],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Feb 21 2013 *)
Table[(Binomial[2 n, n] - Binomial[n, n/2] Cos[Pi n/2])/2, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 04 2016 *)

a(n) = sum(k=0, n, binomial(n, k)^2*(k % 2)); \\ Michel Marcus, Oct 05 2016

a(n) = Sum_{k=0..n} C(n, k)^2*(1-(-1)^k)/2.
a(n) = C(2n-1, n-1)(1-(-1)^n)/2+(C(2n, n)/2-(-1)^(n/2)*C(n, floor(n/2))/2)(1+(-1)^n)/2.
a(n) = (binomial(2*n, n) - binomial(n, n/2)*cos(Pi*n/2))/2 = n^2 * hypergeom([1/2-n/2, 1/2-n/2, 1-n/2, 1-n/2], [1, 3/2, 3/2], 1). - Vladimir Reshetnikov, Oct 04 2016
a(n) = A159916(2n,n). - Alois P. Heinz, Feb 06 2017

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

A123162 Triangle read by rows: T(n,k) = binomial(2n - 1, 2k - 1) for 0 < k <= n and T(n,0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 35, 21, 1, 1, 9, 84, 126, 36, 1, 1, 11, 165, 462, 330, 55, 1, 1, 13, 286, 1287, 1716, 715, 78, 1, 1, 15, 455, 3003, 6435, 5005, 1365, 105, 1, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 1, 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1

Offset: 0

Roger L. Bagula, Oct 02 2006

			Triangle begins:
     1;
     1,  1;
     1,  3,   1;
     1,  5,  10,    1;
     1,  7,  35,   21,    1;
     1,  9,  84,  126,   36,    1;
     1, 11, 165,  462,  330,   55,    1;
     1, 13, 286, 1287, 1716,  715,   78,  1;
     1, 15, 455, 3003, 6435, 5005, 1365, 105, 1;
     ...

Muniru A Asiru, Rows n = 0..50, flattened

Cf. A002458, A007318, A034867, A103327, A122366, A123166, A259557.

Flat(Concatenation([1],List([1..10],n->Concatenation([1],List([1..n],m->Binomial(2*n-1,2*m-1)))))); # Muniru A Asiru, Oct 11 2018

A123162:= func< n,k | k eq 0 select 1 else Binomial(2*n-1, 2*k-1) >;
[A123162(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023

T[n_, k_]= If [k==0, 1, Binomial[2*n-1, 2*k-1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

T(n, k) := if k = 0 then 1 else binomial(2*n - 1, 2*k - 1)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */

def A123162(n,k): return binomial(2*n-1, 2*k-1) + int(k==0)
flatten([[A123162(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2022

From Paul Barry, May 26 2008: (Start)
T(n,k) = binomial(2*n - 1, 2*k - 1) + 0^k.
Column k has g.f. (x^k/(1 - x)^(2*k + 0^k))*Sum_{j=0..k} binomial(2*k, 2*j)*x^j. (End)
From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
Row n = coefficients in the expansion of ((x + sqrt(x))*(sqrt(x) - 1)^(2*n) + (x - sqrt(x))*(sqrt(x) + 1)^(2*n) + 2*x - 2)/(2*x - 2).
G.f.: (1 - (2 + x)*y + (1 - 2*x)*y^2 - (x - x^2)*y^3)/(1 - (3 + 2*x)*y + (3 + x^2)*y^2 - (1 - 2*x + x^2)*y^3).
E.g.f.: ((x + sqrt(x))*exp(y*(sqrt(x) - 1)^2) + (x - sqrt(x))*exp(y*(sqrt(x) + 1)^2) + (2*x - 2)*exp(y) - 2*x)/(2*x - 2). (End)
From G. C. Greubel, Jul 18 2023: (Start)
Sum_{k=0..n} T(n,k) = A123166(n).
T(n, n-1) = (n-1)*T(n, 1), n >= 2.
T(2*n, n) = A259557(n).
T(2*n+1, n+1) = A002458(n). (End)

Edited by N. J. A. Sloane, Oct 04 2006

Partially edited and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018

A337397 Expansion of sqrt(2 / ( (1+64x^2) (1-8x+sqrt(1+64x^2)) )).

Original entry on oeis.org

1, 2, -34, -92, 1654, 4828, -88724, -268088, 4984486, 15361708, -287691196, -898052872, 16901635516, 53234639768, -1005474931816, -3187958034544, 60375963282182, 192405594166988, -3651655920615596, -11684176213422568, 222132094724096852, 713091439789994824, -13575872676384218776

Offset: 0

Seiichi Manyama, Aug 26 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=4 of A337464.

Cf. A002458, A193618, A337396.

a[n_] := Sum[(-4)^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 26 2020 *)
CoefficientList[Series[Sqrt[2/((1+64x^2)(1-8x+Sqrt[1+64x^2]))],{x,0,30}],x] (* Harvey P. Dale, Jul 24 2021 *)

N=40; x='x+O('x^N); Vec(sqrt(2/((1+64*x^2)*(1-8*x+sqrt(1+64*x^2)))))

{a(n) = sum(k=0, n, (-4)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

a(0) = 1, a(1) = 2 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * 2 * a(n-1) - 64 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

A032123 Number of 2n-bead black-white reversible strings with n black beads.

Original entry on oeis.org

1, 1, 4, 10, 38, 126, 472, 1716, 6470, 24310, 92504, 352716, 1352540, 5200300, 20060016, 77558760, 300546630, 1166803110, 4537591960, 17672631900, 68923356788, 269128937220, 1052049834576, 4116715363800, 16123803193628, 63205303218876, 247959271674352, 973469712824056

Offset: 0

Christian G. Bower

nonn

It appears that a(n) is also the number of quivers in the mutation class of affine B_n or affine type C_n for n>=2. [Christian Stump, Nov 02 2010]

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. G. Bower, Transforms (2)
N. J. A. Sloane, Classic Sequences

Central column of Losanitsch's triangle A034851.

Cf. A002458 (bisection).

With[{nn = 50}, CoefficientList[Series[Exp[x]*Cosh[x]*BesselI[0, 2*x], {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, Feb 15 2017 *)

a(2n+1) = binomial(4n+1,2n) = A002458(n). a(2n) = binomial(4n-1,2n-1)+binomial(2n-1,n-1), n>0.
"BIK[ n ](2n-1)" (reversible, indistinct, unlabeled, n parts, 2n-1 elements) transform of 1, 1, 1, 1...
E.g.f.: exp(x)*cosh(x)*BesselI(0, 2*x). - Vladeta Jovovic, Apr 07 2005
G.f.: (1/2)*((1-4*x)^(-1/2)+(1-4*x^2)^(-1/2)). - Mark van Hoeij, Oct 30 2011
Conjecture: D-finite with recurrence n*(n-1)*a(n) -2*(n-1)*(3*n-4)*a(n-1) +4*(2*n^2-14*n+19)*a(n-2) +8*(n^2+5*n-19)*a(n-3) -16*(n-3)*(3*n-10)*a(n-4) +32*(n-4)*(2*n-9)*a(n-5)=0, n>5. - R. J. Mathar, Nov 09 2013
a(n) ~ 2^(2*n-1)/sqrt(Pi*n). - Vaclav Kotesovec, Mar 29 2014

A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2ax^2 + 1)^(n+1) from x = 0 to infinity.

Original entry on oeis.org

1, 6, 4, 42, 60, 24, 308, 688, 560, 160, 2310, 7080, 8760, 5040, 1120, 17556, 68712, 114576, 99456, 44352, 8064, 134596, 642824, 1351840, 1572480, 1055040, 384384, 59136, 1038312, 5864640, 14912064, 21778560, 19536000, 10695168, 3294720

Offset: 0

R. J. Mathar, Mar 17 2007

The integral N(a;n) = Integral_{x=0..infinity} 1/(x^4 + 2*a*x^2 + 1)^(n+1) has a polynomial representation P_n(a) = 2^(n + 3/2) * (a+1)^(n + 1/2) * N(a;n) / Pi (known as the Boros-Moll polynomial). The table contains the coefficients T(n,l) of P_n(a) = 2^(-2*n)*Sum_{l=0..n} T(n,l)*a^l in row n and column l (with n >= 0 and 0 <= l <= n).

			The table T(n,l) (with rows n >= 0 and columns l = 0..n) starts:
      1;
      6,     4;
     42,    60,     24;
    308,   688,    560,   160;
   2310,  7080,   8760,  5040,  1120;
  17556, 68712, 114576, 99456, 44352, 8064;
  ...
For n = 2, N(a;2) = Integral_{x=0..oo} dx/(x^4 + 2*a*x + 1)^3 = 2^(-2*2)*(Sum_{l=0..2} T(2,l)*a^l) * Pi/(2^(2 + 3/2) * (a + 1)^(2 + 1/2) = (42 + 60*a + 24*a^2) * Pi/(32 * (2*(a+1))^(5/2)) for a > -1. - _Petros Hadjicostas_, May 25 2020

Tewodros Amdeberhan and Victor H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, arXiv:0707.2118 [math.CA], 2007.
George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, arXiv:0806.4333 [math.CO], 2009.
William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, Mathematics of Computation, 78(268) (2009), 2269-2282.
Victor H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
Victor H. Moll, Combinatorial sequences arising from a rational integral, Onl. J. Anal. Combin., no 2 (2007), #4.

Cf. A002458 (row sums), A004982 (column l=0), A059304 (main diagonal), A067001 (rows reversed), A223549, A223550, A334907.

A126936 := proc(m, l)
    add(2^k*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m):
end:
seq(seq(A126936(m,l), l=0..m), m=0..12); # R. J. Mathar, May 25 2020

t[m_, l_] := Sum[2^k*Binomial[2*m-2*k, m-k]*Binomial[m+k, m]*Binomial[k, l], {k, l, m}]; Table[t[m, l], {m, 0, 11}, {l, 0, m}] // Flatten (* Jean-François Alcover, Jan 09 2014, after Maple, adapted May 2020 *)

From Petros Hadjicostas, May 25 2020: (Start)
T(n,l) = A067001(n, n-l) = 2^(2*n) * A223549(n,l)/A223550(n,l).
Sum_{l=0..n} T(n,l) = A002458(n) = A334907(n)*2^n/n!.
Bivariate o.g.f.: Sum_{n,l >= 0} T(n,l)*x^n*y^l = sqrt((1 + y)/(1 - 8*x*(1 + y))/(y + sqrt(1 - 8*x*(1 + y)))). (End)

Corrected by Petros Hadjicostas, May 23 2020

A277247 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)^2.

Original entry on oeis.org

1, 1, 5, 10, 53, 126, 662, 1716, 8885, 24310, 124130, 352716, 1778966, 5200300, 25947612, 77558760, 383358645, 1166803110, 5719519850, 17672631900, 85990654178, 269128937220, 1300866635172, 4116715363800, 19780031677718, 63205303218876, 302045506654052

Offset: 0

Vladimir Reshetnikov, Oct 06 2016

Interleaves A036910 and A002458.

Cf. A000984, A002458, A036910, A110145.

A277247 := proc(n)
    add(binomial(n,k)^2,k=0..floor(n/2)) ;
end proc:
seq(A277247(n),n=0..50) ; # R. J. Mathar, Jan 11 2024

Table[(Binomial[2 n, n] + (Binomial[n, n/2] Cos[Pi n/2])^2)/2, {n, 0, 30}]
CoefficientList[Series[(1/Sqrt[1-4x]+(2EllipticK[16 x^2])/Pi)/2, {x, 0, 20}], x] (* Benedict W. J. Irwin, Oct 19 2016 *)

a(n) = (binomial(2*n, n) + (binomial(n, n/2)*cos(Pi*n/2))^2)/2.
D-finite with recurrence: 2*(2*n+1)*(4*n^2+15*n+13)*(16*(n+1)^2*a(n) - (n+2)^2*a(n+2)) = (n+2)*(4*n^2+7*n+2)*(16*(n+2)^2*a(n+1) - (n+3)^2*a(n+3)).
G.f.: (1/sqrt(1 - 4*x) + 2*K(4*x)/Pi)/2, where K is the complete elliptic integral of the first kind with modulus 4*x. - Benedict W. J. Irwin, Oct 19 2016
D-finite with recurrence n^2*(n-1)*a(n) -2*(3*n-4)*(n-1)^2*a(n-1) +4*(-19*n^2+64*n-56)*a(n-2) +16*(4*n^3-11*n^2-16*n+49)*a(n-3) -64*(4*n-15)*(n-3)^2*a(n-4) +256*(2*n-9)*(n-4)^2*a(n-5)=0. - R. J. Mathar, Jan 11 2024

A111505 Right half of Pascal's triangle (A007318) with zeros.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 6, 4, 1, 0, 0, 0, 10, 5, 1, 0, 0, 0, 20, 15, 6, 1, 0, 0, 0, 0, 35, 21, 7, 1, 0, 0, 0, 0, 70, 56, 28, 8, 1, 0, 0, 0, 0, 0, 126, 84, 36, 9, 1, 0, 0, 0, 0, 0, 252, 210, 120, 45, 10, 1, 0, 0, 0, 0, 0, 0, 462, 330, 165

Offset: 0

Philippe Deléham, Nov 16 2005

A034869 is the version without zeros.

			Triangle begins:
1;
0, 1;
0, 2, 1;
0, 0, 3, 1;
0, 0, 6, 4, 1;
0, 0, 0, 10, 5, 1;
0, 0, 0, 20, 15, 6, 1;
0, 0, 0, 0, 35, 21, 7, 1;
0, 0, 0, 0, 70, 56, 28, 8, 1;
0, 0, 0, 0, 0, 126, 84, 36, 9, 1;
0, 0, 0, 0, 0, 252, 210, 120, 45, 10, 1;
0, 0, 0, 0, 0, 0, 462, 330, 165, 55, 11, 1;
0, 0, 0, 0, 0, 0, 924, 792, 495, 220, 66, 12, 1;
0, 0, 0, 0, 0, 0, 0, 1716, 1287, 715, 286, 78, 13, 1;
0, 0, 0, 0, 0, 0, 0, 3432, 3003, 2002, 1001, 364, 91, 14, 1;

Cf. A007318, A034869, A094527, A111418.

Sum_{n, n>=k} T(n, k) = A001700(k).
Sum_{k =0..2*n} T(2*n, k) = A032443(n).
Sum_{k=0..2*n+1} T(2*n+1, k) = 4^n = A000302(n).
Sum_{k=0..2*n} T(2*n, k)^2 = A036910(n).
Sum_{k=0..2*n+1} T(2*n+1, k)^2 = C(4*n+1, 2*n) = A002458(n) . Paul D. Hanna

cp's OEIS Frontend

A100033 Bisection of A001700.

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A110145 a(n) = Sum_{k=0..n} C(n,k)^2*mod(k,2).

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A099976 Bisection of A000984.

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A123162 Triangle read by rows: T(n,k) = binomial(2*n - 1, 2*k - 1) for 0 < k <= n and T(n,0) = 1.

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A337397 Expansion of sqrt(2 / ( (1+64*x^2) * (1-8*x+sqrt(1+64*x^2)) )).

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A032123 Number of 2n-bead black-white reversible strings with n black beads.

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A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2*a*x^2 + 1)^(n+1) from x = 0 to infinity.

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A123162 Triangle read by rows: T(n,k) = binomial(2n - 1, 2k - 1) for 0 < k <= n and T(n,0) = 1.

A337397 Expansion of sqrt(2 / ( (1+64x^2) (1-8x+sqrt(1+64x^2)) )).

A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2ax^2 + 1)^(n+1) from x = 0 to infinity.