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.

Showing 1-3 of 3 results.

A176332 Row sums of triangle A176331.

Original entry on oeis.org

1, 2, 5, 16, 56, 202, 741, 2752, 10318, 38972, 148070, 565280, 2166646, 8332378, 32136205, 124249856, 481433286, 1868972828, 7267804550, 28304698336, 110383060776, 431000853028, 1684754608210, 6592277745536, 25818887839956
Offset: 0

Views

Author

Paul Barry, Apr 15 2010

Keywords

Comments

Hankel transform is A176333.
Let A(n) denote the n X n array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i-1 times to the tuple ((sqrt(-1))^m, 1 <= m <= n). Then the negative of the real part of the (n, n)-entry of A(n) equals a(n-2) for all n > 2. - John M. Campbell, Jan 20 2019

Links

Crossrefs

Cf. A000108, A176331, A176333.
Cf. A005317, A024718, A360185, A360211.

Programs

  • GAP
    List([0..30], n -> Sum([0..n], k -> Sum([0..n], j -> (-1)^(n-j)* Binomial(j, n-k)*Binomial(j, k) ))) # G. C. Greubel, Feb 22 2019
  • Magma
    [(&+[ (&+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]]): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 21 2019
  • Maple
    a:=n->add(add(binomial(j,n-k)*binomial(j,k)*(-1)^(n-j),j=0..n),k=0..n): seq(a(n),n=0..30); # Muniru A Asiru, Jan 23 2019
  • Mathematica
    f[n_]:= (-1)^n*Sum[Binomial[n+k, k] Cos[Pi(n+k)/2], {k, 0, n}]; Array[f, 24, 0] (* Robert G. Wilson v, Apr 02 2012 *)
  • PARI
    {a(n) = sum(k=0,n, sum(j=0,n, (-1)^(n-j)*binomial(j,n-k)* binomial(j,k))) };vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 21 2019
  • PARI
    a(n) = {my(v = vector(n, k, I^k)); for (k=1, n-1, v = vector(n, i, sum(j=1, i, v[j]));); -real(v[n]);} \\ Michel Marcus, Feb 25 2019
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-2*k, n)); \\ Seiichi Manyama, Jan 29 2023
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^2*(2/(1+sqrt(1-4*x)))^2))) \\ Seiichi Manyama, Jan 29 2023
  • Sage
    [sum(sum((-1)^(n-j)*binomial(j,n-k)*binomial(j,k) for j in (0..n)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Feb 21 2019

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(j,n-k)*binomial(j,k)*(-1)^(n-j).
Logarithm g.f.: arctan(x*C(x)) = Sum_{n>=1} a(n)/n*x^n, where C(x) = (1-sqrt(1-4*x))/(2*x) (A000108). - Vladimir Kruchinin, Aug 10 2010
Conjecture: 6*n*a(n) + 2*(-17*n+10)*a(n-1) + (47*n-60)*a(n-2) + 10*(-3*n+5)*a(n-3) + 4*(2*n-5)*a(n-4) = 0. - R. J. Mathar, Nov 24 2012
Recurrence: 2*n*(5*n-8)*a(n) = 2*(25*n^2 - 50*n + 18)*a(n-1) - (45*n^2 - 92*n + 36)*a(n-2) + 2*(2*n-3)*(5*n-3)*a(n-3). - Vaclav Kotesovec, Feb 12 2014
a(n) ~ 4^(n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014
From Seiichi Manyama, Jan 29 2023: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-2*k,n).
G.f.: 1 / ( sqrt(1-4*x) * (1 + x^2 * c(x)^2) ), where c(x) is the g.f. of A000108. (End)
a(n) = [x^n] 1/((1+x^2) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024

A176335 Central coefficients T(2n,n) of number triangle A176331.

Original entry on oeis.org

1, 3, 28, 315, 3876, 50358, 678112, 9365499, 131809060, 1882294128, 27193657008, 396600597198, 5829739893264, 86262567856650, 1283677784658528, 19196304797150715, 288295493121264420, 4346056823245242420
Offset: 0

Views

Author

Paul Barry, Apr 15 2010

Keywords

Links

Crossrefs

Cf. A112029, A176331, A176332, A176334.

Programs

  • GAP
    T:= function(n,k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
  end;
List([0..30], n-> T(2*n,n) ); # G. C. Greubel, Dec 07 2019
  • Magma
    T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
[T(2*n,n): n in [0..30]]; // G. C. Greubel, Dec 07 2019
  • Maple
    A176335 := proc(n)
    add((-1)^k*binomial(k,n)^2,k=0..2*n);
end proc: # R. J. Mathar, Feb 10 2015
  • Mathematica
    T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[T[2*n, n], {n,0,30}] (* G. C. Greubel, Dec 07 2019 *)
  • PARI
    T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
vector(31, n, T(2*(n-1), n-1) ) \\ G. C. Greubel, Dec 07 2019
  • Sage
    @CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[T(2*n, n) for n in (0..30)] # G. C. Greubel, Dec 07 2019

Formula

a(n) = Sum_{k=0..2n} C(k,n)^2*(-1)^k.
Conjecture: 224*n^2*(n-1)*a(n) - 48*(n-1)*(65*n^2 - 36*n - 13)*a(n-1) + 4*(-1839*n^3 + 11081*n^2 - 21932*n + 14280)*a(n-2) + 12*(-81*n^3 + 326*n^2 - 591*n + 562)*a(n-3) - (n-3)*(1853*n^2 - 7403*n + 7140)*a(n-4) - 12*(n-4)*(2*n-7)^2*a(n-5) = 0. - R. J. Mathar, Feb 10 2015
From Peter Bala, Aug 08 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n+k, k)^2. Cf. A112029.
Conjecture (assuming an offset of 1): the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 5 and all positive integers n and r [added Nov 29 2024: proved by Coster. See Theorem 4]. (End)
a(n) ~ 2^(4*n+2) / (5*Pi*n). - Vaclav Kotesovec, Aug 08 2024
a(n) = binomial(2*n, n)^2 * hypergeom([1, -n, -n], [-2*n, -2*n], -1). - Peter Luschny, Nov 29 2024

A176334 Diagonal sums of number triangle A176331.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 124, 305, 755, 1879, 4698, 11792, 29694, 74984, 189811, 481498, 1223713, 3115200, 7942134, 20275362, 51823246, 132604193, 339644739, 870745187, 2234208932, 5737129623, 14742751524, 37909928908, 97543380598
Offset: 0

Views

Author

Paul Barry, Apr 15 2010

Keywords

Links

Crossrefs

Cf. A176331, A176332, A176335.

Programs

  • GAP
    T:= function(n,k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
  end;
List([0..30], n-> Sum([0..Int(n/2)], j-> T(n-j,j) )); # G. C. Greubel, Dec 07 2019
  • Magma
    T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
[(&+[T(n-k,k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Dec 07 2019
  • Maple
    A176334 := proc(n)
    add(add(binomial(j,n-2*k)*binomial(j,k)*(-1)^(n-k-j),j=0..n-k), k=0..floor(n/2)) ;
end proc: # R. J. Mathar, Feb 10 2015
  • Mathematica
    T[n_, k_]:= T[n, k]= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[Sum[T[n-k, k], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Dec 07 2019 *)
  • PARI
    T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
vector(30, n, sum(j=0, (n-1)\2, T(n-j-1,j)) ) \\ G. C. Greubel, Dec 07 2019
  • Sage
    @CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Dec 07 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(j,n-2k)*C(j,k)*(-1)^(n-k-j).
a(n) ~ phi^(2*n+3) / (4*5^(1/4)*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 08 2024
Showing 1-3 of 3 results.

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