A218045 -id:A218045 - cp's OEIS frontend

A288470 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n,2*k).

1, 2, 14, 92, 646, 4652, 34124, 253528, 1901638, 14368844, 109208164, 833981128, 6394017436, 49185717752, 379438594136, 2934361958192, 22741538394694, 176582855512588, 1373431963785332, 10698376362421096, 83447762846703796, 651690159076273192, 5095051571420324264, 39874449115469939152, 312350761370734541596

Offset: 0

Robert Israel, Jun 09 2017

Row sums of A155495.

a(n) is the constant term in the expansion of (-1 + (1 + x + 1/x)^2)^n. - Seiichi Manyama, Nov 21 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1109

Cf. A094061, A155495, A218045.

f:= gfun:-rectoproc({n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1), a(0)=1,a(1)=2, a(2)=14},a(n),remember):
map(f, [$0..30]);

Table[Sum[Binomial[n, k] Binomial[2 n, 2 k], {k, 0, n}], {n, 0, 24}] (* Michael De Vlieger, Jun 09 2017 *)

{a(n) = polcoef((-1+(1+x+1/x)^2)^n, 0)} \\ Seiichi Manyama, Nov 21 2019

a(n) = hypergeom([-n,-n,1/2-n],[1/2,1],-1).
n*(2*n-1)*a(n) = (32*(n-2))*(2*n-5)*a(n-3)+(8*(9*n^2-31*n+28))*a(n-2)+(2*(3*n^2+7*n-9))*a(n-1).
G.f.: sqrt((1-2*x+sqrt(1-8*x))/(2*(1-7*x-8*x^2))).
a(n) ~ 8^n / sqrt(3*Pi*n). - Vaclav Kotesovec, Nov 27 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(3*n-2*k-1,n-2*k). - Seiichi Manyama, Feb 13 2024
From Peter Bala, Aug 30 2025: (Start)
n*(2*n - 1)*(3*n - 4)*a(n) = 2*(21*n^3 - 49*n^2 + 33*n - 6)*a(n-1) + 8*(n - 1)*(3*n - 1)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2.
a(n) = Sum_{0 <= i, j <= n/2} binomial(2*n, j)*binomial(2*n+i-1, i)*binomial(2*n, n- 2*i-2*j) (verified to satisfy the above second-order recurrence using the MulZeil procedure in Doron Zeilberger's MultiZeilberger Maple package).
Equivalently, a(n) = [x^n] ( (1 + x + x^2 + x^3)/(1 - x^2) )^(2*n). Cf. A240688.
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k)  for all primes p and positive integers n and k.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

A369263 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2)^3 ).

Original entry on oeis.org

1, 2, 10, 54, 329, 2126, 14356, 100030, 713956, 5193064, 38354066, 286860714, 2168308302, 16537766036, 127114940840, 983657456878, 7657060437148, 59917814944376, 471062428422152, 3718952705982232, 29471640802526185, 234356062245289566, 1869405604537134116

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A218045, A369208.
Cf. A369262, A369264.
Cf. A370245.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^2)^3)/x)

a(n, s=2, t=3, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(3*n-2*k+1,n-2*k).

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^2)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A378290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n+2r+k,r) binomial(r,n-r)/(n+2*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 19, 0, 1, 4, 15, 46, 104, 0, 1, 5, 22, 82, 262, 614, 0, 1, 6, 30, 128, 486, 1588, 3816, 0, 1, 7, 39, 185, 789, 3027, 10053, 24595, 0, 1, 8, 49, 254, 1185, 5052, 19543, 65686, 162896, 0, 1, 9, 60, 336, 1689, 7801, 33290, 129606, 439658, 1101922, 0

Offset: 0

Seiichi Manyama, Nov 21 2024

			Square array begins:
  1,    1,     1,     1,     1,     1,     1, ...
  0,    1,     2,     3,     4,     5,     6, ...
  0,    4,     9,    15,    22,    30,    39, ...
  0,   19,    46,    82,   128,   185,   254, ...
  0,  104,   262,   486,   789,  1185,  1689, ...
  0,  614,  1588,  3027,  5052,  7801, 11430, ...
  0, 3816, 10053, 19543, 33290, 52490, 78552, ...

Columns k=0..2 give A000007, A186997, A218045(n+2).

Cf. A009766, A026300, A378289, A378291, A378292.

T(n, k, t=3, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + x * A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A186997.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-2,k+3) for n > 1.

A369208 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2) ).

Original entry on oeis.org

1, 2, 8, 38, 200, 1122, 6576, 39790, 246672, 1558658, 10001592, 64997814, 426922392, 2829624514, 18901301984, 127115260894, 859978039840, 5848754717314, 39964745880552, 274231943135686, 1888891689752680, 13055393137141282, 90517646431869328

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A218045, A369263.
Cf. A052709, A109081.
Cf. A370242.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^2))/x)

a(n, s=2, t=1, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(3*n-2*k+1,n-2*k).

a(n) = (1/(n+1)) *[x^n] ( 1/(1-x)^2 * (1+x^2) )^(n+1). - Seiichi Manyama, Feb 14 2024

A369226 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^2 ).

Original entry on oeis.org

1, 1, 4, 13, 53, 220, 968, 4373, 20271, 95705, 458904, 2228220, 10934524, 54143848, 270189008, 1357428997, 6860264323, 34853234867, 177900211204, 911867479717, 4691701977973, 24222505191984, 125448280976224, 651555603531308, 3392951906596708, 17711433386188300

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Cf. A052709, A369262.

Cf. A218045, A370243.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+x^2)^2)/x)

a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

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

a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^2)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024

A218182 Number of truth tables of bracketed formulas (case 1).

Original entry on oeis.org

0, 0, 1, 6, 33, 194, 1198, 7676, 50581, 340682, 2335186, 16237284, 114255994, 812107412, 5822171548, 42052209400, 305714145869, 2235262899418, 16426616425002, 121265916776148, 898878250833358, 6687497426512700, 49920590244564484

Offset: 0

N. J. A. Sloane, Oct 23 2012

nonn

Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv preprint arXiv:1205.5595 [math.CO], 2012.

Cf. A186997, A218045.

my(x='x+O('x^30)); concat([0,0], Vec(((1-8*x)^(1/2)-3)*((2+2*(1-8*x)^(1/2)+8*x)^(1/2)-2)/8)) \\ Michel Marcus, Oct 21 2020

Yildiz gives a g.f.

G.f.: ((1-8*x)^(1/2)-3)*((2+2*(1-8*x)^(1/2)+8*x)^(1/2)-2)/8. - Mark van Hoeij, May 16 2013

cp's OEIS Frontend

A288470 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n,2*k).

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A369263 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2)^3 ).

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A378290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n+2*r+k,r) * binomial(r,n-r)/(n+2*r+k) for k > 0.

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A369208 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2) ).

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A369226 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^2 ).

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A218182 Number of truth tables of bracketed formulas (case 1).

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A288470 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2n,2*k).

A378290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n+2r+k,r) binomial(r,n-r)/(n+2*r+k) for k > 0.